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We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many words which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the (
$\kappa -1$
)-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with
$O(n^{t-\varepsilon })$
many words of length n where
$t = \kappa (\kappa +1)/2$
. We prove a variant of the
$1$
-Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension less than
$1$
.
The level of distribution of a complex-valued sequence $b$ measures the quality of distribution of $b$ along sparse arithmetic progressions $nd+a$. We prove that the Thue–Morse sequence has level of distribution $1$, which is essentially best possible. More precisely, this sequence gives one of the first nontrivial examples of a sequence satisfying a Bombieri–Vinogradov-type theorem for each exponent $\theta <1$. This result improves on the level of distribution $2/3$ obtained by Müllner and the author. As an application of our method, we show that the subsequence of the Thue–Morse sequence indexed by $\lfloor n^c\rfloor$, where $1 < c < 2$, is simply normal. This result improves on the range $1 < c < 3/2$ obtained by Müllner and the author and closes the gap that appeared when Mauduit and Rivat proved (in particular) that the Thue–Morse sequence along the squares is simply normal.
As pointed out by Alexandre Bailleul, the paper mentioned in the title contains a mistake in Theorem 2.2. The hypothesis on the linear relation of the almost periods is not sufficient. In this note, we fix the problem and its minor consequences on other results in the same paper.
Let
$a_1$
,
$a_2$
, and
$a_3$
be distinct reduced residues modulo q satisfying the congruences
$a_1^2 \equiv a_2^2 \equiv a_3^2 \ (\mathrm{mod}\ q)$
. We conditionally derive an asymptotic formula, with an error term that has a power savings in q, for the logarithmic density of the set of real numbers x for which
$\pi (x;q,a_1)> \pi (x;q,a_2) > \pi (x;q,a_3)$
. The relationship among the
$a_i$
allows us to normalize the error terms for the
$\pi (x;q,a_i)$
in an atypical way that creates mutual independence among their distributions, and also allows for a proof technique that uses only elementary tools from probability.
We obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we show that for the one-dimensional torus translation, these estimates are nearly optimal.
We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions
${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$
of
$\unicode[STIX]{x1D6FD}$
-maps
$T_{\unicode[STIX]{x1D6FD}}$
, for arbitrary
$\unicode[STIX]{x1D6FD}>1$
.
Kifer, Peres, and Weiss proved in [A dimension gap for continued fractions with independent digits. Israel J. Math.124 (2001), 61–76] that there exists
$c_{0}>0$
, such that
$\dim \unicode[STIX]{x1D707}\leq 1-c_{0}$
for any probability measure
$\unicode[STIX]{x1D707}$
, which makes the digits of the continued fraction expansion independent and identically distributed random variables. In this paper we prove that amongst this class of measures, there exists one whose dimension is maximal. Our results also apply in the more general setting of countable branched systems.
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 generalize current known distribution results on Shanks–Rényi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let
$\unicode[STIX]{x1D70B}(x;q,a)$
be the number of primes up to
$x$
that are congruent to
$a$
modulo
$q$
. For a fixed integer
$q$
and distinct invertible congruence classes
$a_{0},a_{1},\ldots ,a_{D}$
, assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of real
$x$
for which the inequalities
$\unicode[STIX]{x1D70B}(x;q,a_{0})>\unicode[STIX]{x1D70B}(x;q,a_{1})>\cdots >\unicode[STIX]{x1D70B}(x;q,a_{D})$
are simultaneously satisfied admits a logarithmic density.
Let
$\unicode[STIX]{x1D6FD}>1$
be a real number and define the
$\unicode[STIX]{x1D6FD}$
-transformation on
$[0,1]$
by
$T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$
. Let
$f:[0,1]\rightarrow [0,1]$
and
$g:[0,1]\rightarrow [0,1]$
be two Lipschitz functions. The main result of the paper is the determination of the Hausdorff dimension of the set
where
$\unicode[STIX]{x1D70F}_{1}$
,
$\unicode[STIX]{x1D70F}_{2}$
are two positive continuous functions with
$\unicode[STIX]{x1D70F}_{1}(x)\leq \unicode[STIX]{x1D70F}_{2}(y)$
for all
$x,y\in [0,1]$
.
We determine the order of magnitude of $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, and $0\leqslant q\leqslant 1$. In the Steinhaus case, this is equivalent to determining the order of $\lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}|\sum _{n\leqslant x}n^{-it}|^{2q}\,dt$.
In particular, we find that $\mathbb{E}|\sum _{n\leqslant x}f(n)|\asymp \sqrt{x}/(\log \log x)^{1/4}$. This proves a conjecture of Helson that one should have better than squareroot cancellation in the first moment and disproves counter-conjectures of various other authors. We deduce some consequences for the distribution and large deviations of $\sum _{n\leqslant x}f(n)$.
The proofs develop a connection between $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$ and the $q$th moment of a critical, approximately Gaussian, multiplicative chaos and then establish the required estimates for that. We include some general introductory discussion about critical multiplicative chaos to help readers unfamiliar with that area.
For a one-parameter subgroup action on a finite-volume homogeneous space, we consider the set of points admitting divergent-on-average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension of the homogeneous space. As a corollary we know that the Hausdorff dimension of the set of points admitting divergent trajectories is not full, which proves a conjecture of Cheung [Hausdorff dimension of the set of singular pairs, Ann. of Math. (2) 173 (2011), 127–167].
We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$, where $\unicode[STIX]{x1D6E5}$ is a finite-index subgroup of $\text{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$, where for all $j$, $\mathbf{a}_{j}$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\text{SL}(n+1,\mathbb{R})$ developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.
Let $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ be an ergodic measure-preserving system, let $A\in {\mathcal{B}}$ and let $\unicode[STIX]{x1D716}>0$. We study the largeness of sets of the form
for various families $\{f_{1},\ldots ,f_{k}\}$ of sequences $f_{i}:\mathbb{N}\rightarrow \mathbb{N}$. For $k\leq 3$ and $f_{i}(n)=if(n)$, we show that $S$ has positive density if $f(n)=q(p_{n})$, where $q\in \mathbb{Z}[x]$ satisfies $q(1)$ or $q(-1)=0$ and $p_{n}$ denotes the $n$th prime; or when $f$ is a certain Hardy field sequence. If $T^{q}$ is ergodic for some $q\in \mathbb{N}$, then, for all $r\in \mathbb{Z}$, $S$ is syndetic if $f(n)=qn+r$. For $f_{i}(n)=a_{i}n$, where $a_{i}$ are distinct integers, we show that $S$ can be empty for $k\geq 4$, and, for $k=3$, we found an interesting relation between the largeness of $S$ and the abundance of solutions to certain linear equations in sparse sets of integers. We also provide some partial results when the $f_{i}$ are distinct polynomials.
In this work we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin–Iosifescu asymptotics for the exceedances of digits obtained from the regular continued fraction expansion of a number chosen randomly from $(0,1)$ according to the Gauss measure. As a consequence, we significantly improve the best known upper bound on the rate of convergence of the maxima in this case. We observe that the asymptotics of order statistics and the extremal point process can also be investigated using our methods.
Let
$\unicode[STIX]{x1D6FD}>1$
be an integer or, generally, a Pisot number. Put
$T(x)=\{\unicode[STIX]{x1D6FD}x\}$
on
$[0,1]$
and let
$S:[0,1]\rightarrow [0,1]$
be a piecewise linear transformation whose slopes have the form
$\pm \unicode[STIX]{x1D6FD}^{m}$
with positive integers
$m$
. We give a sufficient condition for
$T$
and
$S$
to have the same generic points. We also give an uncountable family of maps which share the same set of generic points.
We investigate the growth rate of the Birkhoff sums
$S_{n,\unicode[STIX]{x1D6FC}}f(x)=\sum _{k=0}^{n-1}f(x+k\unicode[STIX]{x1D6FC})$
, where
$f$
is a continuous function with zero mean defined on the unit circle
$\mathbb{T}$
and
$(\unicode[STIX]{x1D6FC},x)$
is a ‘typical’ element of
$\mathbb{T}^{2}$
. The answer depends on the meaning given to the word ‘typical’. Part of the work will be done in a more general context.
Let
$\unicode[STIX]{x1D6F9}:[1,\infty )\rightarrow \mathbb{R}_{+}$
be a non-decreasing function,
$a_{n}(x)$
the
$n$
th partial quotient of
$x$
and
$q_{n}(x)$
the denominator of the
$n$
th convergent. The set of
$\unicode[STIX]{x1D6F9}$
-Dirichlet non-improvable numbers,
$$\begin{eqnarray}G(\unicode[STIX]{x1D6F9}):=\{x\in [0,1):a_{n}(x)a_{n+1}(x)>\unicode[STIX]{x1D6F9}(q_{n}(x))\text{ for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$
is related with the classical set of
$1/q^{2}\unicode[STIX]{x1D6F9}(q)$
-approximable numbers
${\mathcal{K}}(\unicode[STIX]{x1D6F9})$
in the sense that
${\mathcal{K}}(3\unicode[STIX]{x1D6F9})\subset G(\unicode[STIX]{x1D6F9})$
. Both of these sets enjoy the same
$s$
-dimensional Hausdorff measure criterion for
$s\in (0,1)$
. We prove that the set
$G(\unicode[STIX]{x1D6F9})\setminus {\mathcal{K}}(3\unicode[STIX]{x1D6F9})$
is uncountable by proving that its Hausdorff dimension is the same as that for the sets
${\mathcal{K}}(\unicode[STIX]{x1D6F9})$
and
$G(\unicode[STIX]{x1D6F9})$
. This gives an affirmative answer to a question raised by Hussain et al [Hausdorff measure of sets of Dirichlet non-improvable numbers. Mathematika64(2) (2018), 502–518].
We prove the Hausdorff measure version of the matrix form of Gallagher’s theorem in the inhomogeneous setting, thereby proving a conjecture posed by Hussain and Simmons [‘The Hausdorff measure version of Gallagher’s theorem—closing the gap and beyond’, J. Number Theory186 (2018), 211–225].
For
$\unicode[STIX]{x1D6FD}\in (1,2]$
the
$\unicode[STIX]{x1D6FD}$
-transformation
$T_{\unicode[STIX]{x1D6FD}}:[0,1)\rightarrow [0,1)$
is defined by
$T_{\unicode[STIX]{x1D6FD}}(x)=\unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$
. For
$t\in [0,1)$
let
$K_{\unicode[STIX]{x1D6FD}}(t)$
be the survivor set of
$T_{\unicode[STIX]{x1D6FD}}$
with hole
$(0,t)$
given by
$$\begin{eqnarray}K_{\unicode[STIX]{x1D6FD}}(t):=\{x\in [0,1):T_{\unicode[STIX]{x1D6FD}}^{n}(x)\not \in (0,t)\text{ for all }n\geq 0\}.\end{eqnarray}$$
In this paper we characterize the bifurcation set
$E_{\unicode[STIX]{x1D6FD}}$
of all parameters
$t\in [0,1)$
for which the set-valued function
$t\mapsto K_{\unicode[STIX]{x1D6FD}}(t)$
is not locally constant. We show that
$E_{\unicode[STIX]{x1D6FD}}$
is a Lebesgue null set of full Hausdorff dimension for all
$\unicode[STIX]{x1D6FD}\in (1,2)$
. We prove that for Lebesgue almost every
$\unicode[STIX]{x1D6FD}\in (1,2)$
the bifurcation set
$E_{\unicode[STIX]{x1D6FD}}$
contains infinitely many isolated points and infinitely many accumulation points arbitrarily close to zero. On the other hand, we show that the set of
$\unicode[STIX]{x1D6FD}\in (1,2)$
for which
$E_{\unicode[STIX]{x1D6FD}}$
contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for
$E_{2}$
, the bifurcation set of the doubling map. Finally, we give for each
$\unicode[STIX]{x1D6FD}\in (1,2)$
a lower and an upper bound for the value
$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$
such that the Hausdorff dimension of
$K_{\unicode[STIX]{x1D6FD}}(t)$
is positive if and only if
$t<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$
. We show that
$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}\leq 1-(1/\unicode[STIX]{x1D6FD})$
for all
$\unicode[STIX]{x1D6FD}\in (1,2)$
.