arXiv subjects

Bence Borda

Publications and source records attributed to Bence Borda.

28 records · Page 2Linked to original sources

arXiv2021

Maximizing Sudler products via Ostrowski expansions and cotangent sums

There is an extensive literature on the asymptotic order of Sudler's trigonometric product $P_N (α) = \prod_{n=1}^N |2 \sin (πn α)|$ for fixed or for "typical" values of $α$. In the present paper we establish a structural result, which for a given $α$ characterizes those $N$ for which $P_N(α)$ attains particularly large values. This characterization relies on the coefficients of $N$ in its Ostrowski expansion with respect to $α$, and allows us to obtain very precise estimates for $\max_{1 \le N \leq M} P_N(α)$ and for $\sum_{N=1}^M P_N(α)^c$ in terms of $M$, for any $c>0$. Furthermore, our arguments give a natural explanation of the fact that the value of the hyperbolic volume of the complement of the figure-eight knot appears generically in results on the asymptotic order of the Sudler product and of the Kashaev invariant.

math.NT↗

arXiv2020

On the discrepancy of random subsequences of $\{nα\}$ II

Let $α$ be an irrational number, let $X_1, X_2, \ldots$ be independent, identically distributed, integer-valued random variables, and put $S_k=\sum_{j=1}^k X_j$. Assuming that $X_1$ has finite variance or heavy tails $P (|X_1|>t)\sim ct^{-β}$, $0<β<2$, in Part I of this paper we proved that, up to logarithmic factors, the order of magnitude of the discrepancy $D_N (S_k α)$ of the first $N$ terms of the sequence $\{S_k α\}$ is $O(N^{-τ})$, where $τ= \min (1/(βγ), 1/2)$ (with $β=2$ in the case of finite variances) and $γ$ is the strong Diophantine type of $α$. This shows a change of behavior of the discrepancy at $βγ=2$. In this paper we determine the exact order of magnitude of $D_N (S_k α)$ for $βγ<1$, and determine the limit distribution of $N^{-1/2} D_N (S_k α)$. We also prove a functional version of these results describing the asymptotic behavior of a wide class of functionals of the sequence $\{S_k α\}$. Finally, we extend our results to the discrepancy of $\{S_k\}$ for general random walks $S_k$ without arithmetic conditions on $X_1$, assuming only a mild polynomial rate on the weak convergence of $\{S_k\}$ to the uniform distribution.

math.PR↗

arXiv2020

Berry-Esseen smoothing inequality for the Wasserstein metric on compact Lie groups

We prove a sharp general inequality estimating the distance of two probability measures on a compact Lie group in the Wasserstein metric in terms of their Fourier transforms. We use a generalized form of the Wasserstein metric, related by Kantorovich duality to the family of functions with an arbitrarily prescribed modulus of continuity. The proof is based on smoothing with a suitable kernel, and a Fourier decay estimate for continuous functions. As a corollary, we show that the rate of convergence of random walks on semisimple groups in the Wasserstein metric is necessarily almost exponential, even without assuming a spectral gap. Applications to equidistribution and empirical measures are also given.

math.CA↗

arXiv2020

Equidistribution of random walks on compact groups II. The Wasserstein metric

We consider a random walk $S_k$ with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum $\sum_{k=1}^N f(S_k)$ with Hölder continuous test functions $f$, including the central limit theorem, the law of the iterated logarithm and an almost sure approximation by a Wiener process, provided the distribution of $S_k$ converges to the Haar measure in the $p$-Wasserstein metric fast enough. As an example we construct discrete random walks on an irrational lattice on the torus $\mathbb{R}^d/\mathbb{Z}^d$, and find their precise rate of convergence to uniformity in the $p$-Wasserstein metric. The proof uses a new Berry--Esseen type inequality for the $p$-Wasserstein metric on the torus, and the simultaneous Diophantine approximation properties of the lattice. These results complement the first part of this paper on random walks with an absolutely continuous component and quantitative ergodic theorems for Borel measurable test functions.

math.PR↗

arXiv2019

On the discrepancy of random subsequences of $\{nα\}$

For irrational $α$, $\{nα\}$ is uniformly distributed mod 1 in the Weyl sense, and the asymptotic behavior of its discrepancy is completely known. In contrast, very few precise results exist for the discrepancy of subsequences $\{n_k α\}$, with the exception of metric results for exponentially growing $(n_k)$. It is therefore natural to consider random $(n_k)$, and in this paper we give nearly optimal bounds for the discrepancy of $\{n_k α\}$ in the case when the gaps $n_{k+1}-n_k$ are independent, identically distributed, integer-valued random variables. As we will see, the discrepancy behavior is determined by a delicate interplay between the distribution of the gaps $n_{k+1}-n_k$ and the rational approximation properties of $α$. We also point out an interesting critical phenomenon, a sudden change of the order of magnitude of the discrepancy of $\{n_k α\}$ as the Diophantine type of $α$ passes through a certain critical value.

math.NT↗

arXiv2019

Equidistribution of random walks on compact groups

Let $X_1, X_2, \dots$ be independent, identically distributed random variables taking values from a compact metrizable group $G$. We prove that the random walk $S_k=X_1 X_2 \cdots X_k$, $k=1,2,\dots$ equidistributes in any given Borel subset of $G$ with probability $1$ if and only if $X_1$ is not supported on any proper closed subgroup of $G$, and $S_k$ has an absolutely continuous component for some $k \ge 1$. More generally, the sum $\sum_{k=1}^N f(S_k)$, where $f:G \to \mathbb{R}$ is Borel measurable, is shown to satisfy the strong law of large numbers and the law of the iterated logarithm. We also prove the central limit theorem with remainder term for the same sum, and construct an almost sure approximation of the process $\sum_{k \le t} f(S_k)$ by a Wiener process provided $S_k$ converges to the Haar measure in the total variation metric.

math.PR↗

arXiv2018

Lattice points in algebraic cross-polytopes and simplices

The number of lattice points $\left| tP \cap \mathbb{Z}^d \right|$, as a function of the real variable $t>1$ is studied, where $P \subset \mathbb{R}^d$ belongs to a special class of algebraic cross-polytopes and simplices. It is shown that the number of lattice points can be approximated by an explicitly given polynomial of $t$ depending only on $P$. The error term is related to a simultaneous Diophantine approximation problem for algebraic numbers, as in Schmidt's theorem. The main ingredients of the proof are a Poisson summation formula for general algebraic polytopes, and a representation of the Fourier transform of the characteristic function of an arbitrary simplex in the form of a complex line integral.

math.NT↗

arXiv2018

Bounded error uniformity of the linear flow on the torus

A linear flow on the torus $\mathbb{R}^d / \mathbb{Z}^d$ is uniformly distributed in the Weyl sense if the direction of the flow has linearly independent coordinates over $\mathbb{Q}$. In this paper we combine Fourier analysis and the subspace theorem of Schmidt to prove bounded error uniformity of linear flows with respect to certain polytopes if, in addition, the coordinates of the direction are all algebraic. In particular, we show that there is no van Aardenne--Ehrenfest type theorem for the mod $1$ discrepancy of continuous curves in any dimension, demonstrating a fundamental difference between continuous and discrete uniform distribution theory.

math.NT↗

arXiv2017

On the distribution of the van der Corput sequence in arbitrary base

A central limit theorem with explicit error bound, and a large deviation result are proved for a sequence of weakly dependent random variables of a special form. As a corollary, under certain conditions on the function $f: [0,1] \to \mathbb{R}$ a central limit theorem and a large deviation result are obtained for the sum $\sum_{n=0}^{N-1} f(x_n)$, where $x_n$ is the base $b$ van der Corput sequence for an arbitrary integer $b \ge 2$. Similar results are also proved for the $L^p$ discrepancy of the same sequence for $1 \le p < \infty$. The main methods used in the proofs are the Berry-Esseen theorem and Fourier analysis.

math.NT↗

arXiv2016

On the theorem of Davenport and generalized Dedekind sums

A symmetrized lattice of $2n$ points in terms of an irrational real number $α$ is considered in the unit square, as in the theorem of Davenport. If $α$ is a quadratic irrational, the square of the $L^2$ discrepancy is found to be $c( α) \log n + O \left( \log \log n \right)$ for a computable positive constant $c( α)$. For the golden ratio $φ$, the value $\sqrt{c (φ) \log n}$ yields the smallest $L^2$ discrepancy of any sequence of explicitly constructed finite point sets in the unit square. If the partial quotients $a_k$ of $α$ grow at most polynomially fast, the $L^2$ discrepancy is found in terms of $a_k$ up to an explicitly bounded error term. It is also shown that certain generalized Dedekind sums can be approximated using the same methods. For a special generalized Dedekind sum with arguments $a, b$ an asymptotic formula in terms of the partial quotients of $\frac{a}{b}$ is proved.

math.NT↗