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Let $p$ be a prime number. For a positive integer $n$ and a real number $\xi$, let $\lambda _n (\xi )$ denote the supremum of the real numbers $\lambda$ for which there are infinitely many integer tuples $(x_0, x_1, \ldots , x_n)$ such that $| x_0 \xi - x_1|_p, \ldots , | x_0 \xi ^{n} - x_n|_p$ are all less than $X^{-\lambda - 1}$, where $X$ is the maximum of $|x_0|, |x_1|, \ldots , |x_n|$. We establish new results on the Hausdorff dimension of the set of real numbers $\xi$ for which $\lambda _n (\xi )$ is equal to (or greater than or equal to) a given value.
Let
$g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let
$\Lambda _0$ be a basic hyperbolic set of the geodesic flow of
$g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of
$g_0$ and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let
$L_{g,\Lambda ,f}$ (respectively
$M_{g,\Lambda ,f}$) be the Lagrange (respectively Markov) spectrum of asymptotic highest (respectively highest) values of f along the geodesics in the hyperbolic continuation
$\Lambda $ of
$\Lambda _0$. We prove that for generic choices of g and f, the Hausdorff dimensions of the sets
$L_{g,\Lambda , f}\cap (-\infty , t)$ vary continuously with
$t\in \mathbb {R}$ and, moreover,
$M_{g,\Lambda , f}\cap (-\infty , t)$ has the same Hausdorff dimension as
$L_{g,\Lambda , f}\cap (-\infty , t)$ for all
$t\in \mathbb {R}$.
We define a family $\mathcal {B}(t)$ of compact subsets of the unit interval which provides a filtration of the set of numbers whose continued fraction expansion has bounded digits. We study how the set $\mathcal {B}(t)$ changes as the parameter t ranges in $[0,1]$, and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behaviour as the family of real quadratic polynomials. The set $\mathcal {E}$ of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension $1$. The Hausdorff dimension of $\mathcal {B}(t)$ varies continuously with the parameter, and we show that the dimension of each individual set equals the dimension of the corresponding section of the bifurcation set $\mathcal {E}$.
We consider the continued fraction expansion of real numbers under the action of a nonuniform lattice in
$\text {PSL}(2,{\mathbb R})$ and prove metric relations between the convergents and a natural geometric notion of good approximations.
We construct a family of fibred threefolds $X_m \to (S , \Delta )$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,\Delta )$ of general type. Following Campana, the threefolds $X_m$ are called weakly special but not special. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if m is big enough, the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conjecture. We prove our results by adapting the recent method of Ru and Vojta. We also formulate some generalisations of known conjectures on exceptional loci that fit into Campana’s program and prove some cases over function fields.
Let
$ (G_n)_{n=0}^{\infty } $
be a nondegenerate linear recurrence sequence whose power sum representation is given by
$ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $
. We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions,
$ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $
for
$ n $
large enough.
Let
${\mathbf {G}}$
be a semisimple algebraic group over a number field K,
$\mathcal {S}$
a finite set of places of K,
$K_{\mathcal {S}}$
the direct product of the completions
$K_{v}, v \in \mathcal {S}$
, and
${\mathcal O}$
the ring of
$\mathcal {S}$
-integers of K. Let
$G = {\mathbf {G}}(K_{\mathcal {S}})$
,
$\Gamma = {\mathbf {G}}({\mathcal O})$
and
$\pi :G \rightarrow G/\Gamma $
the quotient map. We describe the closures of the locally divergent orbits
${T\pi (g)}$
where T is a maximal
$K_{\mathcal {S}}$
-split torus in G. If
$\# S = 2$
then the closure
$\overline {T\pi (g)}$
is a finite union of T-orbits stratified in terms of parabolic subgroups of
${\mathbf {G}} \times {\mathbf {G}}$
and, consequently,
$\overline {T\pi (g)}$
is homogeneous (i.e.
$\overline {T\pi (g)}= H\pi (g)$
for a subgroup H of G) if and only if
${T\pi (g)}$
is closed. On the other hand, if
$\# \mathcal {S}> 2$
and K is not a
$\mathrm {CM}$
-field then
$\overline {T\pi (g)}$
is homogeneous for
${\mathbf {G}} = \mathbf {SL}_{n}$
and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for
${\mathbf {G}} \neq \mathbf {SL}_{n}$
. As an application, we prove that
$\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$
for the class of non-rational locally K-decomposable homogeneous forms
$f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$
.
Let G be a semisimple real algebraic group defined over
${\mathbb {Q}}$
,
$\Gamma $
be an arithmetic subgroup of G, and T be a maximal
${\mathbb {R}}$
-split torus. A trajectory in
$G/\Gamma $
is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which accounts for the divergence. If this is the case, the divergent trajectory is called obvious. Given a closed cone in T, we study the existence of non-obvious divergent trajectories under its action in
$G\kern-1pt{/}\kern-1pt\Gamma $
. We get a sufficient condition for the existence of a non-obvious divergence trajectory in the general case, and a full classification under the assumption that
$\mathrm {rank}_{{\mathbb {Q}}}G=\mathrm {rank}_{{\mathbb {R}}}G=2$
.
Jeśmanowicz conjectured that
$(x,y,z)=(2,2,2)$
is the only positive integer solution of the equation
$(*)\; ((\kern1.5pt f^2-g^2)n)^x+(2fgn)^y=((\kern1.5pt f^2+g^2)n)^x$
, where n is a positive integer and f, g are positive integers such that
$f>g$
,
$\gcd (\kern1.5pt f,g)=1$
and
$f \not \equiv g\pmod 2$
. Using Baker’s method, we prove that: (i) if
$n>1$
,
$f \ge 98$
and
$g=1$
, then
$(*)$
has no positive integer solutions
$(x,y,z)$
with
$x>z>y$
; and (ii) if
$n>1$
,
$f=2^rs^2$
and
$g=1$
, where r, s are positive integers satisfying
$(**)\; 2 \nmid s$
and
$s<2^{r/2}$
, then
$(*)$
has no positive integer solutions
$(x,y,z)$
with
$y>z>x$
. Thus, Jeśmanowicz’ conjecture is true if
$f=2^rs^2$
and
$g=1$
, where r, s are positive integers satisfying
$(**)$
.
We prove that, for any small $\varepsilon > 0$, the number of irrationals among the following odd zeta values: $\zeta (3),\zeta (5),\zeta (7),\ldots ,\zeta (s)$ is at least $( c_0 - \varepsilon )({s^{1/2}}/{(\log s)^{1/2}})$, provided $s$ is a sufficiently large odd integer with respect to $\varepsilon$. The constant $c_0 = 1.192507\ldots$ can be expressed in closed form. Our work improves the lower bound $2^{(1-\varepsilon )({\log s}/{\log \log s})}$ of the previous work of Fischler, Sprang and Zudilin. We follow the same strategy of Fischler, Sprang and Zudilin. The main new ingredient is an asymptotically optimal design for the zeros of the auxiliary rational functions, which relates to the inverse totient problem.
We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis [Logarithm laws for flows on homogeneous spaces. Invent. Math.138(3) (1999), 451–494] resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. Selecta Math.10 (2004), 479–523], hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson–Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW’s sufficient conditions for extremality. In the first of this series of papers [Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. Selecta Math.24(3) (2018), 2165–2206], we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, as well as proving the ‘inherited exponent of irrationality’ version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson–Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying.
Recently E. Bombieri and N. M. Katz (2010) demonstrated that several well-known results about the distribution of values of linear recurrence sequences lead to interesting statements for Frobenius traces of algebraic curves. Here we continue this line of study and establish the Möbius randomness law quantitatively for the normalised form of Frobenius traces.
It is known that the Fourier–Stieltjes coefficients of a nonatomic coin-tossing measure may not vanish at infinity. However, we show that they could vanish at infinity along some integer subsequences, including the sequence
${\{b^{n}\}}_{n\geq 1}$
where
$b$
is multiplicatively independent of 2 and the sequence given by the multiplicative semigroup generated by 3 and 5. The proof is based on elementary combinatorics and lower-bound estimates for linear forms in logarithms from transcendental number theory.
For an irrational number
$x\in [0,1)$
, let
$x=[a_{1}(x),a_{2}(x),\ldots ]$
be its continued fraction expansion with partial quotients
$\{a_{n}(x):n\geq 1\}$
. Given
$\unicode[STIX]{x1D6E9}\in \mathbb{N}$
, for
$n\geq 1$
, the
$n$
th longest block function of
$x$
with respect to
$\unicode[STIX]{x1D6E9}$
is defined by
$L_{n}(x,\unicode[STIX]{x1D6E9})=\max \{k\geq 1:a_{j+1}(x)=\cdots =a_{j+k}(x)=\unicode[STIX]{x1D6E9}~\text{for some}~j~\text{with}~0\leq j\leq n-k\}$
, which represents the length of the longest consecutive sequence whose elements are all
$\unicode[STIX]{x1D6E9}$
from the first
$n$
partial quotients of
$x$
. We consider the growth rate of
$L_{n}(x,\unicode[STIX]{x1D6E9})$
as
$n\rightarrow \infty$
and calculate the Hausdorff dimensions of the level sets and exceptional sets arising from the longest block function.
We establish various new results on a problem proposed by Mahler [Some suggestions for further research. Bull. Aust. Math. Soc.29 (1984), 101–108] concerning rational approximation to fractal sets by rational numbers inside and outside the set in question. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich, and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover, we provide a non-trivial lower bound for the distance of a rational number $p/q$ outside the Cantor middle-third set $C$ to the set $C$, in terms of the denominator $q$. We further discuss patterns of rational numbers in fractal sets. We highlight two of them: firstly, an upper bound for the number of rational (algebraic) numbers in a fractal set up to a given height (and degree) for a wide class of fractal sets; and secondly, we find properties of the denominator structure of rational points in ‘missing-digit’ Cantor sets, generalizing claims of Nagy and Bloshchitsyn.
We present a complete characterisation of the radial asymptotics of degree-one Mahler functions as
$z$
approaches roots of unity of degree
$k^{n}$
, where
$k$
is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over
$\mathbb{C}(z)$
. Finally, we discuss asymptotic bounds towards generic points on the unit circle.