1 Introduction
In 1966, Schmidt [Reference Schmidt13] introduced a two-player game referred to thereafter as Schmidt’s game. Schmidt invented the game primarily as a tool for studying certain sets which arise in number theory and Diophantine approximation theory. Schmidt’s game and other similar games have since become an important tool in number theory, dynamics, and related areas.
Schmidt’s game (defined precisely in Section 2.3) and related games are real games, that is games in which each player plays a “real” (an element of a Polish space: a completely metrizable and separable space). Questions regarding which player, if any, has a winning strategy in various games have been systematically studied over the last century. Games in which one of the players has a winning strategy are said to be determined. The existence of winning strategies often have implications in both set theory and applications to other areas. In fact, the assumption that certain classes of games are determined can have far-reaching structural consequences. One such assumption is the axiom of determinacy,
$\mathsf {AD}$
, which is the statement that all integer games are determined. The axiom of determinacy for real games,
$\mathsf {AD}_{\mathbb R}$
, would immediately imply the determinacy of Schmidt’s game, but it is significantly stronger than
$\mathsf {AD}$
(see Section 2.1 for a more thorough discussion). A natural question is what form of determinacy axiom is necessary to obtain the determinacy of Schmidt’s game. In particular, can one obtain the determinacy of this game from
$\mathsf {AD}$
, or does one need the full strength of
$\mathsf {AD}_{\mathbb R}$
?
Consider the case of the Banach–Mazur game on a Polish space
$(X,d)$
with target set
$T \subseteq X\!$
. Here the players
${\boldsymbol {I}}$
and
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
at each turn n play a real which codes a closed ball
$B(x_n,\rho _n)=\{ y\in X\colon d(x_n,y)\leq \rho _n\}$
. The only “rule” of the game is that the players must play a decreasing sequence of closed balls (that is, the first player to violate this rule loses). If both players follow the rule, then
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
wins iff
$\bigcap _n B(x_n,\rho _n) \cap T \neq \emptyset $
. Although this is a real game, this game is determined for any
$T \subseteq X$
just from
$\mathsf {AD}$
. This follows from the easy fact that the Banach–Mazur game is equivalent to the integer game in which both players play closed balls with “rational centers” (i.e., from a fixed countable dense set) and rational radii.
For Schmidt’s game on a Polish space
$(X,d)$
with target set
$T\subseteq X\!$
, we have in addition fixed parameters
$\alpha ,\beta \in (0,1)$
. In this game
${\boldsymbol {I}}$
’s first move is a closed ball
$B(x_0,\rho _0)$
as in the Banach–Mazur game. In subsequent moves, the players play a decreasing sequence of closed balls as in the Banach–Mazur game, but with a restriction of the radii. Namely,
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
must shrink the previous radius by a factor of
$\alpha $
, and
${\boldsymbol {I}}$
must shrink the previous radius by
$\beta $
. So, at move
${2n}$
,
${\boldsymbol {I}}$
plays a closed ball of radius
$\rho _{2n}=(\alpha \beta )^n \rho _0$
, and at move
${2n+1}$
,
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
plays a closed ball of radius
$\rho _{2n+1}=\alpha (\alpha \beta )^n \rho _0$
. As with the Banach–Mazur game, if both players follow these rules, then
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
wins iff
$x \in T$
where
$\{ x\}=\bigcap _n B(x_n,\rho _n)$
. We call this game the
$(\alpha ,\beta )$
Schmidt’s game for T. A variation of Schmidt’s game, first introduced by Akhunzhanov in [Reference Akhunzhanov1], has an additional rule that the initial radius
$\rho _0=\rho $
of
${\boldsymbol {I}}$
’s first move is fixed in advance. We call this the
$(\alpha ,\beta ,\rho )$
Schmidt’s game for T. In all practical applications of the game we are aware of, the difference between these two versions is immaterial. However, in general, these games are not literally equivalent, as the following simple example demonstrates.
Example 1.1. Consider
$\mathbb {R}$
with the usual metric and let the target set for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
be
$T=(-\infty , -1] \cup [1, \infty ) \cup \mathbb {Q}$
. Notice that this set is dense. It is easy to see that if
$\rho \geq 2$
and
$\alpha \leq \frac 14$
then for any
$\beta $
,
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
wins the
$(\alpha , \beta , \rho )$
-game, simply by maximizing the distance from the center of her first move to the origin. But if
${\boldsymbol {I}}$
is allowed to choose any starting radius and
$\beta < \frac 12$
, then he is allowed to play, for instance,
$(0, \frac 12)$
, and then on subsequent moves, simply avoid each rational one at a time, so that in fact
${\boldsymbol {I}}$
wins the
$(\alpha , \beta )$
-game.
In the case of Schmidt’s game (either variation) it is not immediately clear that the game is equivalent to an integer game, and thus it is not clear that
$\mathsf {AD}$
suffices for the determinacy of these games. Our main results have implications regarding the determinacy of Schmidt’s game.
Another class of games which is similar in spirit to Schmidt’s game are the so-called Banach games whose determinacy has been investigated by Becker [Reference Becker2] and Freiling [Reference Freiling3] (with an important result being obtained by Martin). Work of these authors has shown that the determinacy of these games follows from (and is, in fact, equivalent to)
$\mathsf {AD}$
. Methods similar to those used by Becker, Freiling, and Martin are instrumental in the proofs of our results as well.
In Section 2 we introduce notation and give some relevant background in the theory of games, descriptive set theory, and the history of Schmidt’s game in particular.
In Section 3 we prove our main results, including those regarding the determinacy of Schmidt’s game. We prove general results, Theorems 3.6 and 3.8, which give some conditions under which certain real games are determined under
$\mathsf {AD}$
alone. Roughly speaking, these results state that “intersection” games which admit strategies which are simple enough to be “coded by a real,” in a sense to made precise, are determined from
$\mathsf {AD}$
. Schmidt’s game, Banach–Mazur games, and other similar games are intersection games. The simple strategy condition, however, depends on the specific game. For Schmidt’s
$(\alpha ,\beta ,\rho )$
game on
$\mathbb R$
, we show the simple strategy condition is met, and so this game is determined from
$\mathsf {AD}$
. Moreover, for the
$(\alpha ,\beta )$
Schmidt’s game on
$\mathbb R$
,
$\mathsf {AD}$
implies that either player
${\boldsymbol {I}}$
has a winning strategy or else for every
$\rho $
,
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
has a winning strategy in the
$(\alpha ,\beta ,\rho )$
game (this does not immediately give a strategy for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
in the
$(\alpha ,\beta )$
game from
$\mathsf {AD}$
, as we are unable in the second case to choose, as a function of
$\rho $
, a winning strategy for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
in the
$(\alpha ,\beta ,\rho )$
game). For
$\mathbb R^n$
,
$n \geq 2$
, the simple strategy condition is not met. In fact, for
$n \geq 3$
we show that the determinacy of Schmidt’s
$(\alpha ,\beta ,\rho )$
games does not follow from
$\mathsf {AD}$
. For
$n=2$
, we do not know if
$\mathsf {AD}$
suffices to get the determinacy of Schmidt’s game.
We end Section 3 by giving an interesting application of the simple strategy hypothesis for Schmidt’s game on
$\mathbb {R}$
to show that whichever player has a winning strategy must have a winning positional strategy i.e., a strategy which needs only the latest move to compute a response. Schmidt [Reference Schmidt13] proved this fact for general intersection games, but the proof heavily relies on the axiom of choice, which we are able to avoid here using simple strategies. Precise statements are included in the section.
In Section 4 we prove two other results related to the determinacy of Schmidt’s game in particular. First, we show assuming
$\mathsf {AD}$
that in any Polish space
$(X,d)$
, any
$p \in (0,1)$
, and any
$T \subseteq X\!$
, there is at most one value of
$(\alpha ,\beta )\in (0,1)^2$
with
$\alpha \beta =p$
such that the
$(\alpha ,\beta )$
Schmidt’s game for T is not determined. Second, we show assuming
$\mathsf {AD}$
that for a general Polish space
$(X,d)$
and any target set
$T\subseteq X\!$
, the “non-tangent” version of Schmidt’s
$(\alpha ,\beta ,\rho )$
game is determined. This game is just like Schmidt’s game except we require each player to play a “non-tangent ball,” that is,
$d(x_n,x_{n+1}) < \rho _n-\rho _{n+1}$
. These results help to illuminate the obstacles in analyzing the determinacy of Schmidt’s game.
Finally in Section 5 we list several open questions which are left unanswered by our results. We feel that the results and questions of the current paper show an interesting interplay between determinacy axioms and the combinatorics of Schmidt’s game.
2 Background
In this section we fix the notation we use to describe the games we will be considering, both for general games and specifically for Schmidt’s game. We recall some facts about the forms of determinacy we will be considering, some necessary background in descriptive set theory to state and prove our theorems, and we explain some of the history and significance of Schmidt’s game.
Throughout we let
$\omega =\mathbb N=\{ 0,1,2,\dots \}$
denote the set of natural numbers. We let
$\mathbb R$
denote the set of real numbers (here we mean the elements of the standard real line, not the Baire space
$\omega ^\omega $
as is frequently customary in descriptive set theory).
2.1 Games
Let X be a non-empty set. Let
$X^{<\omega }$
and
$X^\omega $
denote respectively the set of finite and infinite sequences from X. For
$s \in X^{<\omega }$
we let
$|s|$
denote the length of s. If
$s,t \in X^{<\omega }$
we write
$s \leq t$
if s is an initial segment of t, that is,
$t\restriction |s|=s$
. If
$s,t \in X^{<\omega }$
, we let
$s {}^{\smallfrown } t$
denote the concatenation of s and t.
We call
$R\subseteq X^{<\omega }$
a tree on X if it is closed under initial segments, that is, if
$t \in R$
and
$s\leq t$
, then
$s \in R$
. We can view R as the set of rules for a game. That is, each player must move at each turn so that the finite sequence produced stays in R (the first player to violate this “rule” loses the game). If
$\vec {x}=(x_0,x_1,\dots ) \in X^\omega $
, we say
$\vec {x}$
has followed the rules if
$\vec {x}\restriction n \in R$
for all n. We let
$[R]$
denote the set of all
$\vec x \in X^{\omega }$
such that
$\vec x \restriction n \in R$
for all n (i.e.,
$\vec x$
has followed the rules). We also refer to
$[R$
] as the set of branches through R. We likewise say
$s \in X^{<\omega }$
has followed the rules just to mean
$s \in R$
.
Fix a set
$B \subseteq X^\omega $
, which we call the target set, and let
$R \subseteq X^{<\omega }$
be a rule set (i.e., a tree on X). The game
$G(B,R)$
on the set X is defined as follows.
${\boldsymbol {I}}$
and
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
alternate playing elements
$x_i \in X\!$
. So,
${\boldsymbol {I}}$
plays
$x_0,x_2,\dots $
, while
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
plays
$x_1,x_3,\dots $
. This produces the run of the game
$\vec x=(x_0,x_1,\dots )$
. The first player, if any, to violate the rules R loses the run
$\vec x$
of the game. If both players follow the rules (i.e.,
$\vec x\in [R]$
), then we declare
${\boldsymbol {I}}$
to have won the run iff
$\vec x \in B$
(otherwise we say
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
has won the run). Oftentimes, in defining a game the set of rules R is defined implicitly by giving requirements on each players’ moves. If there are no rules, i.e.,
$R=X^{<\omega }$
, then we write
$G(B)$
for
$G(B,R)$
. Also, it is frequently convenient to define the game by describing the payoff set for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
instead of
${\boldsymbol {I}}\!$
. This, of course, is formally just replacing B with
$X^\omega -B$
.
A strategy for
${\boldsymbol {I}}$
in a game on the set X is a function
$\sigma \colon \bigcup _{n\in \omega } X^{2n} \to X\!$
. A strategy for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
is a function
$\tau \colon \bigcup _{n \in \omega } X^{2n+1}\to X\!$
. We say
$\sigma $
follows the rule set R is whenever
$s \in R$
of even length, than
$s {}^{\smallfrown } \sigma (s)\in R$
. We likewise define the notion of a strategy
$\tau $
for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
to follow the rules. We say
$\vec x \in X^{\omega }$
follows the strategy
$\sigma $
for
${\boldsymbol {I}}$
if for all
$n \in \omega $
,
$x_{2n}=\sigma (\vec x\restriction 2n)$
, and similarly define the notion of
$\vec x$
following the strategy
$\tau $
for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}\!$
. We also extend this terminology in the obvious way to say an
$s \in X^{<\omega }$
has followed
$\sigma $
(or
$\tau $
). Finally, we say a strategy
$\sigma $
for
${\boldsymbol {I}}$
is a winning strategy for
${\boldsymbol {I}}$
in the game
$G(B,R)$
if
$\sigma $
follows the rules R and for all
$\vec x\in [R]$
which follows
$\sigma $
we have
$\vec x\in B$
, that is, player
${\boldsymbol {I}}$
has won the run
$\vec x$
. We likewise define the notion of
$\tau $
being a winning strategy for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}\!$
.
If
$\sigma $
is a strategy for
${\boldsymbol {I}}\!$
, and
$\vec z=(x_1,x_3,\dots )$
is a sequence of moves for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}\!$
, we write
$\sigma * \vec {z} $
to denote the corresponding run
$(x_0,x_1,x_2,x_3,\dots )$
where
$x_{2n}=\sigma ( x\restriction 2n)$
. We likewise define
$\tau * \vec z$
for
$\tau $
a strategy for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
and
$\vec z=(x_0,x_2,\dots )$
a sequence of moves for
${\boldsymbol {I}}\!$
. If
$\sigma , \tau $
are strategies for
${\boldsymbol {I}}$
and
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
respectively, then we let
$\sigma *\tau $
denote the run
$(x_0,x_1,\dots )$
where
$x_{2n}=\sigma ( x\restriction 2n)$
and
$x_{2n+1}=\tau ( x\restriction 2n+1)$
for all n.
We say the game
$G(B,R)$
on X is determined if one of the players has a winning strategy. The axiom of determinacy for games on X, denoted
$\mathsf {AD}_X$
is the assertion that all games on the set X are determined. Axioms of this kind were first introduced by Mycielski and Steinhaus. We let
$\mathsf {AD}$
denote
$\mathsf {AD}_\omega $
, that is, the assertion all two-player integer games are determined. Also important for the current paper is the axiom
$\mathsf {AD}_{\mathbb R}$
, the assertion that all real games are determined. Both
$\mathsf {AD}$
and
$\mathsf {AD}_{\mathbb R}$
play an important role in modern descriptive set theory. Although both axioms contradict the axiom of choice,
$\mathsf {AC}$
, and thus are not adopted as axioms for the true universe V of set theory, they play a critical role in developing the theory of natural models such as
$L(\mathbb {R})$
containing “definable” sets of reals. It is known that
$\mathsf {AD}_{\mathbb R}$
is a much stronger assertion than
$\mathsf {AD}$
(see Theorem 4.4 of [Reference Solovay, Kechris and Moschovakis14]).
Sitting between
$\mathsf {AD}$
and
$\mathsf {AD}_{\mathbb R}$
is the determinacy of another class of games called
$\frac {1}{2}\mathbb R$
games, in which one of the players plays reals and the other plays integers. The proof of one of our theorems will require the use of
$\frac {1}{2}\mathbb R$
games. The axiom
$\mathsf {AD}_{\frac {1}{2}\mathbb R}$
that all
$\frac {1}{2}\mathbb R$
games are determined is known to be equivalent to
$\mathsf {AD}_{\mathbb R}$
(
$\mathsf {AD}_{\frac {1}{2}\mathbb R}$
immediately implies Uniformization; see Theorem 2.3). However,
$\mathsf {AD}$
suffices to obtain the determinacy of
$\frac {1}{2}\mathbb R$
games with Suslin, co-Suslin payoff (a result of Woodin; see [Reference Kechris, Kechris, Martin and Steel4]). We define these terms more precisely in Section 3. As in [Reference Becker2], this fact will play an important role in one of our theorems.
One of the central results in the theory of games is the result of Martin [Reference Martin, Nerode and Shore6] that all Borel games on any set X are determined in
$\mathsf {ZFC}$
. By “Borel” here we are referring to the topology on
$X^\omega $
given by the product of the discrete topologies on X. In fact, in just
$\mathsf {ZF}$
we have that all Borel games (on any set X) are quasi-determined (see [Reference Moschovakis12] for the definition of quasi-strategy and proof of the extension of Martin’s result to quasi-strategies in
$\mathsf {ZF}$
, which is due to Hurkens and Neeman).
Theorem 2.1 (Martin, Hurkens, and Neeman for quasi-strategies).
Let X be a nonempty set, and let
$B\subseteq X^\omega $
be a Borel set, and
$R\subseteq X^{<\omega }$
a rule set R (a tree). Then the game
$G(B,R)$
is determined (assuming
$\mathsf {ZFC}$
, or quasi-determined just assuming
$\mathsf {ZF}$
).
As we mentioned above,
$\mathsf {AD}$
contradicts
$\mathsf {AC}$
. In fact, games played for particular types of “pathological” sets constructed using
$\mathsf {AC}$
are frequently not determined. For example, the following result is well-known (e.g., [Reference Kechris5, p. 137, paragraph 8]):
Proposition 2.2. Let
$B \subseteq \omega ^\omega $
be a Bernstein set (i.e., neither the set nor its complement contains a perfect set). Then the game
$G(B)$
is not determined.
2.2 Determinacy and pointclasses
We briefly review some of the terminology and results related to the determinacy of games and some associated notions concerning pointclasses which we will need for the proofs of some of our results.
We have introduced above the axioms
$\mathsf {AD}$
,
$\mathsf {AD}_{\frac {1}{2}\mathbb R}$
, and
$\mathsf {AD}_{\mathbb R}$
which assert the determinacy of integer games, half-real games, and real games respectively. We trivially have
$\mathsf {AD}_{\mathbb R} \Rightarrow \mathsf {AD}_{\frac {1}{2}\mathbb R} \Rightarrow \mathsf {AD}$
. All three of these axioms contradict
$\mathsf {AC}$
, the axiom of choice. They are consistent, however, with
$\mathsf {DC}$
, the axiom of dependent choice, which asserts that if T is a non-empty pruned tree (i.e., if
$(x_0,\dots ,x_n)\in T$
then
$\exists x_{n+1}\ (x_0,\dots ,x_n,x_{n+1})\in T$
) then there is a branch f through T (i.e.,
$\forall n\ (f(0),\dots ,f(n))\in T$
).
$\mathsf {DC}$
is a slight strengthening of the axiom of countable choice. On the one hand,
$\mathsf {DC}$
holds in the minimal model
$L(\mathbb {R})$
of
$\mathsf {AD}$
, while on the other hand even
$\mathsf {AD}_{\mathbb R}$
does not imply
$\mathsf {DC}$
. Throughout this paper, our background theory is
$\mathsf {ZF}+\mathsf {DC}$
.
The axiom
$\mathsf {AD}_{\mathbb R}$
is strictly stronger than
$\mathsf {AD}$
(see [Reference Solovay, Kechris and Moschovakis14]), and in fact it is known that
$\mathsf {AD}_{\mathbb R}$
is equivalent to
$\mathsf {AD}+\mathrm {Unif}$
, where
$\mathrm {Unif}$
is the axiom that every
$R\subseteq \mathbb R\times \mathbb R$
has a uniformization, that is, a function
$f \colon \mathrm {dom}(R)\to \mathbb R$
such that
$(x,f(x))\in R$
for all
$x \in \mathrm {dom}(R)$
(see Theorem 2.3). This equivalence will be important for our argument in Theorem 3.11 that
$\mathsf {AD}$
does not suffice for the determinacy of Schmidt’s game in
$\mathbb R^n$
for
$n \geq 3$
. The notion of uniformization is closely connected with the descriptive set theoretic notion of a scale. If a set
$R\subseteq X\times Y$
(where X, Y are Polish spaces) has a scale, then it has a uniformization. The only property of scales which we use is the existence of uniformizations, so we will not give the definition, which is rather technical, here (though they are equivalent to Suslin representations, defined below).
A (boldface) pointclass
$\boldsymbol {\Gamma }$
is a collection of subsets of Polish spaces closed under continuous preimages, that is, if
$f \colon X\to Y$
is continuous and
$A\subseteq Y$
is in
$\boldsymbol {\Gamma }$
, then
$f^{-1}(A)$
is also in
$\boldsymbol {\Gamma }$
. We say
$\boldsymbol {\Gamma }$
is self-dual if
$\boldsymbol {\Gamma }=\check {\boldsymbol {\Gamma }}$
where
$\check {\boldsymbol {\Gamma }}=\{ X\setminus A\colon A\in \boldsymbol {\Gamma }\}$
is the dual pointclass of
$\boldsymbol {\Gamma }$
. We say
$\boldsymbol {\Gamma }$
is non-self-dual if
$\boldsymbol {\Gamma }\neq \check {\boldsymbol {\Gamma }}$
. A set
$U \subseteq {\omega ^\omega } \times X$
is universal for the
$\boldsymbol {\Gamma }$
subsets of X if
$U\in \boldsymbol {\Gamma }$
and for every
$A\subseteq X$
with
$A\in \boldsymbol {\Gamma }$
there is an
$x \in {\omega ^\omega }$
with
$A=U_x=\{ y\colon (x,y)\in U\}$
. A standard fact is that for any Polish space X, the usual non-self-dual Borel levels
$\boldsymbol {\Sigma }^0_\alpha \restriction X\!$
,
$\boldsymbol {\Pi }^0_\alpha \restriction X$
are pointclasses and have universal sets, as are the projective levels
$\boldsymbol {\Sigma }^1_n\restriction X\!$
,
$\boldsymbol {\Pi }^1_n \restriction X\!$
. In the case
$X={\omega ^\omega }$
, there is a complete analysis of pointclasses under
$\mathsf {AD}$
. The pointclasses
$\boldsymbol {\Gamma }\restriction {\omega ^\omega }$
fall into a natural well-ordered hierarchy (modulo considering
$\boldsymbol {\Gamma }$
and its dual class
$\check {\boldsymbol {\Gamma }}$
at the same level) by Wadge’s lemma. Furthermore, every non-self-dual
$\boldsymbol {\Gamma }\restriction {\omega ^\omega }$
has a universal set, an easy consequence of Wadge’s lemma (see Theorem 7D.3 of [Reference Moschovakis12]).
In this paper we will be looking at more general pointclasses in more general Polish spaces (in particular in
$\mathbb R^n$
). We note that the general definition of pointclass given above allows in this context some unnatural examples. For example, consider
$\boldsymbol {\Gamma }$
defined by:
$A \in \boldsymbol {\Gamma }\restriction X$
iff
$A\in \boldsymbol {\Pi }^0_{10} \restriction X$
and
$A\cap C$
is closed for every connected component C of the space X (here
$\boldsymbol {\Pi }^0_{10}$
can be replaced by any pointclass). Then it is easy to check that
$\boldsymbol {\Gamma }$
is a pointclass, but
$\boldsymbol {\Gamma }\restriction {\omega ^\omega }= \boldsymbol {\Pi }^0_{10}\restriction {\omega ^\omega }$
and
$\boldsymbol {\Gamma }\restriction \mathbb R= \boldsymbol {\Pi }^0_1\restriction \mathbb R$
. Nevertheless, an arbitrary pointclass
$\boldsymbol {\Gamma }$
in the Baire space (i.e.,
$\boldsymbol {\Gamma }$
is closed under continuous preimages by functions
$f\colon {\omega ^\omega } \to {\omega ^\omega }$
) can be extended to general Polish spaces in a natural way as follows. Given
$\boldsymbol {\Gamma }\restriction {\omega ^\omega }$
, say
$A\in \boldsymbol {\Gamma }'\restriction X$
if for any continuous
$f \colon {\omega ^\omega } \to X$
we have that
$f^{-1}(A)\in \boldsymbol {\Gamma }\restriction {\omega ^\omega }$
. Extended this way, it is easy to check that
$\boldsymbol {\Gamma }'$
is a pointclass and
$\boldsymbol {\Gamma }'\restriction {\omega ^\omega }=\boldsymbol {\Gamma }\restriction {\omega ^\omega }$
. We will henceforth just write
$\boldsymbol {\Gamma }\restriction X$
instead of
$\boldsymbol {\Gamma }'\restriction X$
for this extension. Note that if
$\boldsymbol {\Gamma }$
is a general pointclass (closed under inverse images by continuous functions) then if we consider
$\boldsymbol {\Gamma }\restriction {\omega ^\omega }$
then the extension
$(\boldsymbol {\Gamma } \restriction {\omega ^\omega })'$
of
$\boldsymbol {\Gamma }\restriction {\omega ^\omega }$
to all Polish spaces contains
$\boldsymbol {\Gamma }$
.
Suppose
$\boldsymbol {\Gamma }$
is a pointclass in
${\omega ^\omega }$
which is non-self-dual, and closed under inverse images by
$\boldsymbol {\Sigma }^0_2$
-measurable functions (recall
$f\colon X\to Y$
is
$\boldsymbol {\Sigma }^0_2$
-measurable if
$f^{-1}(U) \in \boldsymbol {\Sigma }^0_2 \restriction X$
for every U open in Y). Then
$\boldsymbol {\Gamma }\restriction X$
has universal sets for all Polish spaces X. This includes all Levy classes, that is pointclasses
$\boldsymbol {\Gamma }$
closed under
$\wedge $
,
$\vee $
, and either
$\exists ^{\omega ^\omega }$
or
$\forall ^{\omega ^\omega }$
. To see this, first note that since
$\boldsymbol {\Gamma }$
is a non-self-dual pointclass in
${\omega ^\omega }$
, it has a universal set
$U\subseteq {\omega ^\omega } \times {\omega ^\omega }$
. Let
$\varphi \colon F\to X$
be one-to-one, onto, continuous, and with
$\varphi ^{-1}$
being
$\boldsymbol {\Sigma }^0_2$
-measurable, where
$F\subseteq {\omega ^\omega }$
is closed (see Theorem 1G.2 of [Reference Moschovakis12] or Theorem 7.9 of [Reference Kechris5]). Let
$\tilde {U}\subseteq {\omega ^\omega } \times X$
be defined by
$\tilde {U}(x,y) \leftrightarrow U(x,\varphi ^{-1}(y))$
. It is straightforward to check that every
$\boldsymbol {\Gamma }\restriction X$
set occurs as a section of
$\tilde {U}$
[If
$A \subseteq X$
is in
$\boldsymbol {\Gamma }$
, let
$r \colon {\omega ^\omega } \to F$
be a continuous retraction (i.e., f is continuous, onto, and
$r\restriction F$
is the identity). Then
$(\varphi \circ r)^{-1}(A)\in \boldsymbol {\Gamma }\restriction {\omega ^\omega }$
. Let x be such that
$U_x= (\varphi \circ r)^{-1}(A)$
. Then
$A=\tilde {U}_x$
.] Also,
$\tilde {U}\in \boldsymbol {\Gamma } \restriction {\omega ^\omega } \times X$
since if
$\psi \colon {\omega ^\omega }\to {\omega ^\omega }\times X$
is continuous, then
$\psi ^{-1}(\tilde {U})(z)\leftrightarrow U(\psi (z)_0, \varphi ^{-1}\circ \psi (z)_1)$
and the function
$z\mapsto (\psi (z)_0,\varphi ^{-1}\circ \psi (z)_1) $
from
${\omega ^\omega }$
to
${\omega ^\omega } \times {\omega ^\omega }$
is
$\boldsymbol {\Sigma }^0_2$
-measurable (here
$\psi (z)=(\psi (z)_0,\psi (z)_1)$
).
For
$\kappa $
an ordinal number we say a set
$A \subseteq {\omega ^\omega }$
is
$\kappa $
-Suslin if there is a tree T on
$\omega \times \kappa $
such that
$A=p[T]$
, where
$p[T]=\{ x \in {\omega ^\omega } \colon \exists f \in \kappa ^\omega \ (x,f)\in [T]\}$
denotes the projection of the body of the tree T. We say A is Suslin if it is
$\kappa $
-Suslin for some
$\kappa $
. We say A is co-Suslin if
${\omega ^\omega } \setminus A$
is Suslin. For a general Polish space X, we say
$A \subseteq X$
is Suslin if for some continuous surjection
$\varphi \colon {\omega ^\omega } \to X$
we have that
$\varphi ^{-1}(A)$
is Suslin (this does not depend on the choice of
$\varphi $
). Scales are essentially the same thing as Suslin representations, in particular a set
$A\subseteq Y$
is Suslin iff it has a scale, thus relations which are Suslin have uniformizations. If
$\boldsymbol {\Gamma }$
is a pointclass, then we say a set A is projective over
$\boldsymbol {\Gamma }$
if it is in the smallest pointclass
$\boldsymbol {\Gamma }'$
containing
$\boldsymbol {\Gamma }$
and closed under complements and existential and universal quantification over
$\mathbb R$
. Assuming
$\mathsf {AD}$
, if
$\boldsymbol {\Gamma }$
is contained in the class of Suslin, co-Suslin sets, then every set projective over
$\boldsymbol {\Gamma }$
is also Suslin and co-Suslin. For this result, more background on these general concepts, as well as the precise definitions of scale and the scale property, the reader can refer to [Reference Moschovakis12].
Results of Martin and Woodin (see [Reference Martin, Kechris, Löwe and Steel7, Reference Martin, Woodin, Kechris, Löwe and Steel9]) show that assuming
$\mathsf {AD}+\mathsf {DC}$
, the axioms
$\mathsf {AD}_{\mathbb R}$
,
$\mathrm {Unif}$
, and scales are all equivalent. More precisely we have the following.
Theorem 2.3 (Martin and Woodin).
Assume
$\mathsf {ZF}+\mathsf {AD}+\mathsf {DC}$
. Then the following are equivalent:
-
(1)
$\mathsf {AD}_{\mathbb R}$
, -
(2)
$\mathrm {Unif}$
, -
(3) Every
$A\subseteq \mathbb R$
has a scale.
Scales and Suslin representations are also important as it follows from
$\mathsf {AD}$
that ordinal games where the payoff set is Suslin and co-Suslin (the notion of Suslin extends naturally to sets
$A \subseteq \lambda ^\omega $
for
$\lambda $
an ordinal number) are determined. One proof of this is due to Moschovakis, Theorem 2.2 of [Reference Moschovakis, Kechris, Martin and Moschovakis11], another due to Steel can be found in the proof of Theorem 2 of [Reference Martin, Steel, Kechris, Martin and Moschovakis8]. We will not need this result for the current paper.
A strengthening of
$\mathsf {AD}$
, due to Woodin, is the axiom
$\mathsf {AD}^+$
. This axiom has been very useful as it allows the development of a structural theory which has been used to obtain a number of results. It is not currently known if
$\mathsf {AD}^+$
is strictly stronger than
$\mathsf {AD}$
, but it holds in all the natural models of
$\mathsf {AD}$
obtained from large cardinal axioms. In particular, if the model
$L(\mathbb {R})$
satisfies
$\mathsf {AD}$
(which it does if there is any inner model containing the reals which satisfies
$\mathsf {AD}$
), then it satisfies
$\mathsf {AD}^+$
. Also, if
$L(\mathbb {R})$
satisfies
$\mathsf {AD}$
, then
$L(\mathbb {R})$
does not satisfy uniformization, and so
$L(\mathbb {R})$
does not satisfy
$\mathsf {AD}_{\mathbb R}$
. So,
$\mathsf {AD}^+$
is strictly weaker than
$\mathsf {AD}_{\mathbb R}$
. In our Theorem 3.11 we in fact show that
$\mathsf {AD}^+$
does not suffice to get the determinacy of Schmidt’s
$(\alpha ,\beta ,\rho )$
game in
$\mathbb R^n$
for
$n \geq 3$
.
2.3 Schmidt’s game
As mentioned in the introduction, Schmidt invented the game primarily as a tool for studying certain sets which arise in number theory and Diophantine approximation theory. These sets are often exceptional with respect to both measure and category, i.e., Lebesgue null and meager. One of the most significant examples is the following. Let
$\mathbb {Q}$
denote the set of rational numbers. A real number x is said to be badly approximable if there exists a positive constant
$c=c(\alpha )$
such that
$\left |x-\frac {p}{q}\right |>\frac {c}{q^2}$
for all
$\frac {p}{q}\in \mathbb {Q}$
. We denote the set of badly approximable numbers by BA. This set plays a major role in Diophantine approximation theory, and is well known to be both Lebesgue null and meager. Nonetheless, using his game, Schmidt was able to prove the following remarkable result:
Theorem 2.4 (Schmidt [Reference Schmidt13]).
Let
$(f_n)_{n=1}^{\infty }$
be a sequence of
$\mathcal C^1$
diffeomorphisms of
$\mathbb R$
. Then the Hausdorff dimension of the set
$\bigcap _{n=1}^{\infty }f^{-1}_n(\mathbf {BA})$
is
$1$
. In particular,
$\bigcap _{n=1}^{\infty }f^{-1}_n(\mathbf {BA})$
is uncountable.
Yet another example of the strength of the game is the following. Let
$b\geq 2$
be an integer. A real number x is said to be normal to base b if, for every
$n\in \mathbb {N}$
, every block of n digits from
$\{0, 1,\dots , b-1\}$
occurs in the base-b expansion of x with asymptotic frequency
$1/b^n$
. It is readily seen that the set of numbers normal to no base is both Lebesgue null and meager. Nevertheless, Schmidt used his game to prove:
Theorem 2.5 (Schmidt [Reference Schmidt13]).
The Hausdorff dimension of the set of numbers normal to no base is
$1$
.
2.3.1 The game’s description
For the
$(\alpha ,\beta )$
Schmidt’s game on the complete metric space
$(X, d)$
with target set
$T \subseteq X\!$
,
${\boldsymbol {I}}$
and
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
each play pairs
$(x_i, \rho _i)$
in
$Y=X \times \mathbb R^{>0}$
. The
$R \subseteq Y^{<\omega }$
of rules is defined by the conditions that
$\rho _{i+1}+d(x_i,x_{i+1})\le \rho _{i}$
and
$\rho _{i+1}= \begin {cases} \alpha \rho _i, & \mathrm { if }\ i\ \mathrm { is\ even },\\ \beta \rho _i, & \mathrm { if }\ i\ \mathrm { is\ odd }.\end {cases}$
The rules guarantee that the closed balls
$B(x_i,\rho _i)=\{ x \in \mathbb R^n \colon d(x,x_i)\leq \rho _i\}$
are nested. Since the
$\rho _i\to 0$
, there is a unique point
$z \in X$
such that
$\{ z\}=\bigcap _i B(x_i,\rho _i)$
. For
$\vec x\in [R]$
, a run of the game following the rules, we let
$f(\vec x)$
be this corresponding point z. The payoff set
$B\subseteq Y^\omega $
for player
${\boldsymbol {I}}$
is
$\{ \vec x \in Y^\omega \cap [R]\colon f(\vec x) \notin T\}$
. Formally, when we refer to the
$(\alpha ,\beta )$
Schmidt’s game with target set T, we are referring to the game
$G(B,R)$
with these sets B and R just described. The formal definition of Schmidt’s
$(\alpha ,\beta ,\rho )$
game with target set T and initial radius
$\rho $
(i.e.,
$\rho _0=\rho $
) is defined in the obvious analogous manner.
3 Main results
We next prove a general result which states that certain real games are equivalent to
$\frac {1}{2}\mathbb R$
games. The essential point is that real games which are intersection games (i.e., games where the payoff only depends on the intersection of sets coded by the moves the players make) with the property that if one of the players has a winning strategy in the real game, then that player has a strategy “coded by a real” (in a precise sense defined below), then the game is equivalent to a
$\frac {1}{2}\mathbb R$
game. In [Reference Becker2] a result attributed to Martin is presented which showed that the determinacy of a certain class of real games, called Banach games, follows from
$\mathsf {AD}_{\frac {1}{2}\mathbb R}$
, the axiom which asserts the determinacy of
$\frac {1}{2}\mathbb R$
games (that is, games in which one player plays reals, and the other plays integers). In Theorem 3.6 we use ideas similar to Martin’s to prove a general result which applies to intersection games satisfying a “simple strategy” hypothesis. Since many games with applications to number theory and dynamics are intersection games, it seems that in practice the simple strategy hypothesis is the more significant requirement.
Definition 3.1. Let
$\boldsymbol {\Gamma }$
be a pointclass. A simple one-round
$\boldsymbol {\Gamma }$
strategy s for the Polish space X is a sequence
$s=(A_n, y_n)_{n \in \omega }$
where
$y_n \in X\!$
,
$A_n \in \boldsymbol {\Gamma }$
, and the
$A_n$
are a partition of X. A simple
$\boldsymbol {\Gamma }$
strategy
$\tau $
for player
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
is a collection
$\{ s_u \}_{u \in \omega ^{<\omega }}$
of simple one-round
$\boldsymbol {\Gamma }$
strategies
$s_u$
. A simple
$\boldsymbol {\Gamma }$
strategy
$\sigma $
for player
${\boldsymbol {I}}$
is a pair
$\sigma =(\bar {y}, \tau )$
where
$\bar {y}\in X$
is the first move and
$\tau $
is a simple
$\boldsymbol {\Gamma }$
strategy for player
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}\!$
.
The idea for a simple one-round strategy is that if the opponent moves in the set
$A_n$
, then the strategy will respond with
$y_n$
. Thus there is only “countably much” information in the strategy; it is coded by a real in a simple manner. If
$s=(A_n,y_n)$
is a simple one-round strategy, we will write
$s(n)=y_n$
and also
$s(x)=y_n$
for any
$x \in A_n$
. A general simple strategy produces after each round a new simple one-round strategy to follow in the next round. For example, suppose
$\sigma $
is a simple strategy for
${\boldsymbol {I}}\!$
.
$\sigma $
gives a first move
$x_0=\bar {y}$
and a simple one-round strategy
$s_\emptyset $
. If
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
plays
$x_1$
, then
$x_2=\sigma (x_0,x_1)=s_\emptyset (x_1)= $
the unique
$y_{n_0}$
such that
$x_1 \in A_{n_0}$
where
$s_\emptyset =(A_n,y_n)$
. If
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
then plays
$x_3$
, then
$\sigma $
responds with
$s_{n_0}(x_3)$
. The play by
$\sigma $
continues in this manner. Formally, a general simple strategy is a sequence
$(s_u)_{u \in \omega ^{<\omega }}$
of simple one-round strategies, indexed by
$u \in \omega ^{<\omega }$
.
If
$\boldsymbol {\Gamma }$
is a pointclass with a universal set
$U\subseteq {\omega ^\omega }\times X\!$
, then we may use U to code simple one-round
$\boldsymbol {\Gamma }$
strategies. Namely, the simple one-round
$\boldsymbol {\Gamma }$
strategy
$s=(A_n,y_n)$
is coded by
$z \in {\omega ^\omega }$
if z codes a sequence
$(z)_n \in {\omega ^\omega }$
and
$U_{(z)_{2n}} =A_n$
and
$(z)_{2n+1}$
codes the response
$y_n\in X$
in some reasonable manner (e.g., via a continuous surjection from
${\omega ^\omega }$
to X, the exact details are unimportant).
Remark 3.2. For the remainder of this section, X and Y will denote Polish spaces.
Definition 3.3. Let
$R \subseteq X^{<\omega }$
be a tree on X which we identify as a set of rules for a game on X. We say a simple one-round
$\boldsymbol {\Gamma }$
strategy s follows the rules R at position
$p \in R$
if for any
$x \in X\!$
, if
$p {}^{\smallfrown } x \in R$
, then
$p {}^{\smallfrown } x {}^{\smallfrown } s(x)\in R$
.
Definition 3.4. Let
$R \subseteq X^{<\omega }$
be a set of rules for a real game. Suppose
$p \in X^{<\omega }$
is a position in R. Suppose
$f \colon X \to X$
is such that for all
$x \in X\!$
, if
$p{}^{\smallfrown } x\in R$
, then
$p{}^{\smallfrown } x {}^{\smallfrown } f(x) \in R$
(i.e., f is a one-round strategy which follows the rules at p). A simplification of f at p is simple one-round strategy
$s=(A_n,y_n)$
such that:
-
(1) For every x in any
$A_n$
, if
$p {}^{\smallfrown } x \in R$
, then
$p {}^{\smallfrown } x {}^{\smallfrown } y_n \in R$
. -
(2) For every n, if there is an
$x \in A_n$
such that
$p{}^{\smallfrown } x \in R$
, then there is an
$x' \in A_n$
with
$p{}^{\smallfrown } x'\in R$
and
$f(x')=y_n$
.
We say s is a
$\boldsymbol {\Gamma }$
simplification of f if all of the sets
$A_n$
are in
$\boldsymbol {\Gamma }$
.
Definition 3.5. We say a tree
$R\subseteq X^{<\omega }$
is positional if for all
$p,q \in R$
of the same length and
$x\in X\!$
, if
$p {}^{\smallfrown } x$
,
$q {}^{\smallfrown } x$
are both in R then for all
$r \in X^{<\omega }$
,
$p{}^{\smallfrown } x{}^{\smallfrown } r \in R$
iff
$q {}^{\smallfrown } x {}^{\smallfrown } r \in R$
.
Theorem 3.6 (
$\mathsf {ZF}+\mathsf {DC}$
).
Let
$\boldsymbol {\Gamma }$
be a pointclass with a universal set with
$\boldsymbol {\Gamma }$
contained within the Suslin, co-Suslin sets. Suppose
$B\subseteq X^\omega $
and
$R \subseteq X^{<\omega }$
is a positional tree, and suppose both B and R are in
$\boldsymbol {\Gamma }$
. Let
$G=G(B,R)$
be the real game on X with payoff B and rules R. Suppose the following two conditions on G hold:
-
(1) (intersection condition) For any
$\vec {x},\vec {y}\in [R]$
, if
$x(2k)=y(2k)$
for all k, then
$\vec {x}\in B$
iff
$\vec {y}\in B$
. -
(2) (simple one-round strategy condition) If
$p\in R$
has odd length, and
$f\colon X \to X$
is a rule following one-round strategy at p, then there is a
$\boldsymbol {\Gamma }$
-simplification of f at p.
Then G is equivalent to a Suslin, co-Suslin
$\frac {1}{2}\mathbb R$
game
$G^*$
in the sense that if
${\boldsymbol {I}}$
(or
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
) has a winning strategy in
$G^*$
, then
${\boldsymbol {I}}$
(or
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
) has a winning strategy in G.
Proof Consider the game
$G^*$
where
${\boldsymbol {I}}$
plays pairs
$(x_{2k},s_{2k})$
and
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
plays integers
$n_{2k+1}$
. The rules
$R^*$
of
$G^*$
are that
${\boldsymbol {I}}$
must play at each round a real coding
$s_{2k}$
which is a simple one-round
$\boldsymbol {\Gamma }$
strategy which follows the rules R relative to a position
$p{}^{\smallfrown } x_{2k}$
for any p of length
$2k$
(this does not depend on the particular choice of p as R is positional).
${\boldsymbol {I}}$
must also play such that
$x_{2k}= s_{2k-2}(n_{2k-1})$
.
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
must play each
$n_{2k+1}$
so that there is a legal move
$x_{2k+1} \in A^{s_{2k}}_{n_{2k+1}}$
with
$p{}^{\smallfrown } x_{2k} {}^{\smallfrown } x_{2k+1} \in R$
(for any p of length
$2k$
).
If
${\boldsymbol {I}}$
and
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
have followed the rules, to produce
$x_{2k}, s_{2k}$
and
$n_{2k+1}$
, the payoff condition for
$G^*$
is as follows. Since
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
has followed the rules, there is a sequence
$x_{2k+1}$
such that the play
$(x_0,x_1,\dots )\in [R]$
.
${\boldsymbol {I}}$
then wins the run of
$G^*$
iff
$(x_0,x_1,\dots )\in B$
. Note that by the intersection condition, this is independent of the particular choice of the
$x_{2k+1}$
.
From the definition,
$G^*$
is a Suslin, co-Suslin game.
We show that
$G^*$
is equivalent to G. Suppose first that
${\boldsymbol {I}}$
wins
$G^*$
by
$\sigma ^*$
. Then
$\sigma ^*$
easily gives a strategy
$\Sigma $
for G. For example, let
$\sigma ^*(\emptyset )= (x_0, s_0)$
. Then
$\Sigma (\emptyset )=x_0$
. If
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
plays
$x_1$
, then let
$n_1$
be such that
$x_1 \in A^{s_0}_{n_1}$
. Then
$\Sigma (x_0,x_1)=s_0(n_1)$
. Continuing in this manner defines
$\Sigma $
. If
$(x_0,x_1,\dots )$
is a run of
$\Sigma $
, then there is a corresponding run
$((x_0,s_0), n_1, \dots )$
of
$\sigma ^*$
. As each
$s_{2k}$
follows the rules R, then as long as
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
’s moves follow the rules R,
${\boldsymbol {I}}$
’s moves by
$\Sigma $
also follow the rules R. If
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
has followed the rules R in the run of G, then the run
$((x_0,s_0), n_1, \dots )$
of
$\sigma ^*$
has followed the rules for
$G^*$
(
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
has followed the rules of
$G^*$
since for each
$n_{2k+1}$
,
$x_{2k+1}$
witnesses that
$n_{2k+1}$
is a legal move). Since
$\sigma ^*$
is winning for
$G^*$
, the sequence
$(x_0,x^{\prime }_1,x_2,x^{\prime }_3,\dots )\in B\cap [R]$
for some
$x^{\prime }_{2k+1}$
. By the intersection condition,
$(x_0,x_1,x_2,x_3,\dots )\in B$
.
Assume now that
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
has winning strategy
$\tau '$
in
$G^*$
. We first note that there is winning strategy
$\tau ^*$
for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
in
$G^*$
such that
$\tau ^*$
is projective over
$\boldsymbol {\Gamma }$
. To see this, first note that the payoff set for
$G^*$
is projective over
$\boldsymbol {\Gamma }$
as both B and R are in
$\boldsymbol {\Gamma }$
. Also, there is a scaled pointclass
$\boldsymbol {\Gamma }'$
, projective over
$\boldsymbol {\Gamma }$
, which contains the payoff set for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
in
$G^*$
. By a result of Woodin in [Reference Kechris, Kechris, Martin and Steel4] (since
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
is playing the integer moves in
$G^*$
) there is a winning strategy
$\tau ^*$
which is projective over
$\boldsymbol {\Gamma }'$
, and thus projective over
$\boldsymbol {\Gamma }$
. For the rest of the proof we fix a winning strategy
$\tau ^*$
for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
in
$G^*$
which is projective over
$\boldsymbol {\Gamma }$
.
We define a strategy
$\Sigma $
for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
in G. Consider the first round of G. Suppose
${\boldsymbol {I}}$
moves with
$x_0$
in G. We may assume that
$(x_0)\in R$
.
Claim 3.7. There is an
$x_1$
with
$(x_0,x_1)\in R$
such that for all
$x_2$
with
$(x_0,x_1,x_2)\in R$
, there is a simple one-round
$\boldsymbol {\Gamma }$
strategy
$s_0$
which follows the rules R from position
$x_0$
(so
$(x_0,s_0)$
is a legal move for
${\boldsymbol {I}}$
in
$G^*$
) such that if
$n_1=\tau ^*(x_0,s_0)$
then
$x_1\in A^{s_0}_{n_1}$
and
$x_2=s_0(x_1)$
.
Proof Suppose not, then for every
$x_1$
with
$(x_0,x_1)\in R$
there is an
$x_2$
with
$(x_0,x_1,x_2)\in R$
which witnesses the failure of the claim. Define the relation
$S(x_1,x_2)$
to hold iff
$(x_0,x_1)\notin R$
or
$(x_0,x_1,x_2)\in R$
and the claim fails, that is, for every simple one-round
$\boldsymbol {\Gamma }$
strategy s which follows R, if we let
$n_1=\tau ^*(x_0,s)$
, then either
$x_1\notin A^{s}_{n_1}$
or
$x_2\neq s(x_1)$
. Since
$\tau ^*$
, B, R are projective over
$\boldsymbol {\Gamma }$
, so is the relation S. By assumption,
$\mathrm {dom}(S)=\mathbb R$
. Since S is projective over
$\boldsymbol {\Gamma }$
, it is within the scaled pointclasses, and thus there is a uniformization f for S. Note that f follows the rules R. By the simple one-round strategy hypothesis of Theorem 3.6, there is a
$\boldsymbol {\Gamma }$
-simplification
$s_0$
of f. Let
$n_1= \tau ^*(x_0,s_0)$
. Since
$\tau ^*$
follows the rules
$R^*$
for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}\!$
, there is an
$x_1 \in A^{s_0}_{n_1}$
such that
$(x_0,x_1)\in R$
. Since
$s_0$
is a simplification of f, there is an
$x^{\prime }_1$
with
$(x_0,x^{\prime }_1)\in R$
and
$f(x^{\prime }_1)=s_0(n_1)$
. Let
$x_2= f(x^{\prime }_1)$
. From the definition of S we have that
$(x_0,x^{\prime }_1,x_2)\in R$
. Since
$S(x^{\prime }_1,x_2)$
, there does not exist an s (following the rules) such that
$(x^{\prime }_1 \in A^{s}_{n_1} ~\mathrm {and}~ x_2=s(x^{\prime }_1))$
where
$n_1=\tau ^*(x_0,s)$
. But on the other hand, the
$s_0$
we have produced does have this property. This proves the claim.
Now that we’ve proved this claim, we can attempt to define the strategy
$\Sigma $
. We would like to have
$\Sigma (x_0)$
be any
$x_1$
as in the claim. Now since the relation
$A(x_0,x_1)$
which says that
$x_1$
satisfies the claim relative to
$x_0$
is projective over
$\boldsymbol {\Gamma }$
, we can uniformize it to produce the first round
$x_1(x_0)$
of the strategy
$\Sigma $
.
Suppose
${\boldsymbol {I}}$
now moves
$x_2$
in G. For each such
$x_2$
such that
$(x_0,x_1,x_2)\in R$
, there is a rule-following simple one-round
$\boldsymbol {\Gamma }$
strategy
$s_0$
as in the claim for
$x_1$
and
$x_2$
. The relation
$A'(x_0,x_2,s_0)$
which says that
$s_0$
satisfies the claim for
$x_1=x_1(x_0)$
,
$x_2$
is projective over
$\boldsymbol {\Gamma }$
and so has a uniformization
$g(x_0,x_2)$
. In the
$G^*$
game we have
${\boldsymbol {I}}$
play
$(x_0, g(x_0,x_2))$
. Note that
$n_1=\tau ^*(x_0,s_0)$
is such that
$x_1 \in A^{s_0}_{n_1}$
, and
$x_2=s_0(x_1)$
.
This completes the definition of the first round of
$\Sigma $
, and the proof that a one-round play according to
$\Sigma $
has a one-round simulation according to
$\tau ^*$
, which will guarantee that
$\Sigma $
wins. The definition of
$\Sigma $
for the general round is defined in exactly the same way, using
$\mathsf {DC}$
to continue. The above argument also shows that a run of G following
$\Sigma $
has a corresponding run of
$G^*$
following
$\tau ^*$
. If
${\boldsymbol {I}}$
has followed the rules of G, then
${\boldsymbol {I}}$
has followed the rules of
$G^*$
in the associated run. Since
$\tau ^*$
is winning for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
in
$G^*$
, there is no sequence
$x^{\prime }_{2k+1}$
of moves for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
such that
$(x_0,x^{\prime }_1,x_2,x^{\prime }_3,\dots )\in B\cap [R]$
. In particular,
$(x_0,x_1,x_2,x_3,\dots ) \notin B$
(since
$(x_0,x_1,\dots )\in [R]$
). Thus,
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
has won the run of G following
$\Sigma $
.
If G is a real game on the Polish space X with rule set R, we say that G is an intersection game if it satisfies the intersection condition of Theorem 3.6. This is equivalent to saying that there is a function
$f\colon X^\omega \to Y$
for some Polish space Y such that
$f(\vec x)=f(\vec y)$
if
$x(2k)=y(2k)$
for all k, and the payoff set for G is of the form
$f^{-1}(T)$
for some
$T\subseteq Y$
. In many examples, the rules R require the players to play decreasing closed sets with diameters going to
$0$
in some Polish space, and the function f is simply giving the unique point of intersection of these sets. If we have a fixed rule set R and a fixed function f, the class of games
$G_{R,f}$
associated with R and f is the collection of games with rules R and payoffs of the form
$f^{-1}(T)$
for
$T\subseteq Y$
. Thus, we allow the payoff set T to vary, but the set of rules R and the “intersection function” f are fixed. In practice, R and f are usually simple, such as Borel relations/functions.
Theorem 3.8 (
$\mathsf {AD}$
).
Suppose
$\boldsymbol {\Gamma }$
is a pointclass within the Suslin, co-Suslin sets and
$G_{R,f}$
is a class of intersection games on the Polish space X with R,
$f \in \boldsymbol {\Gamma }$
, and R is positional (as above
$f \colon X^\omega \to Y$
, where Y is a Polish space). Suppose that for every
$T\subseteq Y$
which is Suslin and co-Suslin, if player
${\boldsymbol {I}}$
or
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
has a winning strategy in
$G_{R,f}(T)$
, then that player has a winning simple
$\boldsymbol {\Gamma }$
-strategy. Then for every
$T\subseteq Y$
, the game
$G_{R,f}(T)$
is determined.
Proof First consider
$\boldsymbol {\Gamma }\restriction {\omega ^\omega }$
. Considering this as a pointclass in the Baire space, there is a larger pointclass
$\boldsymbol {\Gamma }' \restriction {\omega ^\omega }$
which is non-self-dual and closed under
$\wedge , \vee $
, and
$\exists ^{\omega ^\omega }$
and is still within the Suslin and co-Suslin sets. We now extend
$\boldsymbol {\Gamma }'\restriction {\omega ^\omega }$
to all Polish spaces to get
$\boldsymbol {\Gamma }'\restriction X$
as defined in the introduction, and as noted there this extension contains the original
$\boldsymbol {\Gamma } \restriction X\!$
. The closure properties of
$\boldsymbol {\Gamma }'$
ensure that it is closed under substitutions by Borel functions and so (as discussed in the introduction)
$\boldsymbol {\Gamma }'$
has universal sets. So, without loss of generality we may assume that
$\boldsymbol {\Gamma }$
has universal sets.
Fix the rule set R and function f in
$\boldsymbol {\Gamma }$
. Let
$T \subseteq Y$
, and we show the real game
$G_{R,f}(T)$
is determined. Following Becker, we consider the integer game G where
${\boldsymbol {I}}$
and
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
play out reals x and y which code trees (indexed by
$\omega ^{<\omega }$
) of simple one-round
$\boldsymbol {\Gamma }$
strategies. The winning condition for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
is as follows. If exactly one of x, y fails to be a simple
$\boldsymbol {\Gamma }$
-strategy, then that player loses. If both fail to code simple
$\boldsymbol {\Gamma }$
-strategies, then
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
wins. If x codes a simple
$\boldsymbol {\Gamma }$
-strategy
$\sigma _x$
and y codes a simple
$\boldsymbol {\Gamma }$
-strategy
$\tau _y$
, then
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
wins iff
$\sigma _x*\tau _y \in G_{R,f}(T)$
, where
$\sigma *\tau $
denotes the unique sequence of reals obtained by playing
$\sigma $
and
$\tau $
against each other. From
$\mathsf {AD}$
, the game G is determined. Without loss of generality we may assume that
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
has a winning strategy w for G. Let
$S_1\subseteq {\omega ^\omega }$
be the set of z such that z codes a simple
$\boldsymbol {\Gamma }$
-strategy for player
${\boldsymbol {I}}$
which follows the rules R. Likewise,
$S_2$
is the set of z coding rule following
$\boldsymbol {\Gamma }$
-strategies
$\tau _z$
for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}\!$
. Note that
$S_1$
,
$S_2$
are projective over
$\boldsymbol {\Gamma }$
. Let
$$\begin{align*}A=\{ \vec{y} \in X^\omega \colon \exists z \in S_1\ \vec y= \sigma_z * \tau_{w(z)} \}. \end{align*}$$
Since w is a winning strategy for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
in G,
$A\subseteq X^\omega \setminus G_{R,f}(T)$
, so
$f(A) \subseteq Y \setminus T$
. Note that A is projective over
$\boldsymbol {\Gamma }$
by the complexity assumption on R and the fact that
$S_1$
is also projective over
$\boldsymbol {\Gamma }$
. We claim that it suffices to show that
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
wins the real game
$G_{R,f}(Y \setminus f(A))$
. This is because if
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
wins
$G_{R,f}(Y \setminus f(A))$
with run
$\vec y$
, i.e.,
$\vec y \not \in G_{R,f}(Y \setminus f(A))$
, then
$f(\vec y) \in f(A) \subseteq Y \setminus T$
, so
$\vec y \not \in G_{R, f}(T)$
, thus
$\vec y$
is a winning run for
${\boldsymbol {I}\kern -0.05cm\boldsymbol {I}}$
in
$G_{R, f}(T)$
.
We see that
$Y \setminus f(A)$
is projective over
$\boldsymbol {\Gamma }$
, and thus
$G_{R, f}(Y \setminus f(A))$
is equivalent to a Suslin, co-Suslin
$\frac {1}{2}\mathbb R$
game by Theorem 3.6 which is determined (see [Reference Kechris, Kechris, Martin and Steel4]), and so
$G_{R, f}(Y \setminus f(A))$
is determined. Now it suffices to show that
${\boldsymbol {I}}$
doesn’t have a winning strategy in
$G_{R, f}(Y \setminus f(A))$
.
Suppose
${\boldsymbol {I}}$
had a winning strategy for
$G_{R, f}(Y \setminus f(A))$
. By hypothesis,
${\boldsymbol {I}}$
has a winning simple
$\boldsymbol {\Gamma }$
-strategy coded by some
$z \in {\omega ^\omega }$
. Let
$\vec y= \sigma _z* \tau _{w(z)}$
(note that
$z \in S_1$
and so
$w(z)\in S_2$
). Since
$\sigma _z$
is a winning strategy for
${\boldsymbol {I}}$
in
$G_{R, f}(Y \setminus f(A))$
, we have
$f(\vec y) \in Y \setminus f(A)$
. On the other hand, from the definition of A from w we have that
$f(\vec y) \in f(A)$
, a contradiction.
We next apply Theorem 3.8 to deduce the determinacy of Schmidt’s
$(\alpha ,\beta ,\rho )$
games in
$\mathbb R$
from
$\mathsf {AD}$
.
Theorem 3.9 (
$\mathsf {AD}$
).
For any
$\alpha ,\beta \in (0,1)$
, any
$\rho \in \mathbb R_{>0}$
, and any
$T\subseteq \mathbb R$
, the
$(\alpha ,\beta ,\rho )$
Schmidt’s game with target set T is determined.
Proof Let
$\boldsymbol {\Gamma }$
be the pointclass
$\boldsymbol {\Pi }^1_1$
of co-analytic sets. Let R be the tree described by the rules of the
$(\alpha ,\beta ,\rho )$
Schmidt’s game. R is clearly a closed set and is positional. The function f of Theorem 3.8 is given by
$\{ f((x_i,\rho _i)_i) \}= \bigcap _i B(x_i,\rho _i)$
. This clearly satisfies the intersection condition, that is,