arXiv subjects

Andriy Bondarenko

Publications and source records attributed to Andriy Bondarenko.

At least 19 recordsLinked to original sources

arXiv2026

Fourier-invariant functions with dense zero sets

For every $0\leq\beta\leq1/2$, we construct a nonzero real-valued continuous function $f_\beta$ in $L^1(\mathbb R)\cap L^2(\mathbb R)$ such that $\widehat {f}_\beta=f_\beta$ and $f_\beta(\sqrt{n}/[\log(e+n)]^{\beta})=0$ for all $n\geq 0$. The case $\beta=0$ settles in the negative a question raised by Radchenko and Viazovska regarding their Fourier interpolation formula. The construction uses a scale of reproducing kernel Hilbert spaces generated by the Fourier-invariant Hermite functions. Applying the Mehler formula, we identify the reproducing kernels of these spaces. By suitable estimates of these kernels, we show that $(\sqrt{n}/[\log(e+n)]^{\beta})$, with one auxiliary point added to it, is a universal interpolating sequence for at least one of the Hilbert spaces under consideration. However, this result fails when $\beta>1/2$.

math.CA↗

arXiv2026

Siegel zeros and small gaps between zeros of the Riemann zeta function

On assuming the Riemann Hypothesis, we show that Siegel zeros imply the existence of gaps between the zeros of the Riemann zeta function less than 1/2 the normalised length. Specifically, we show that an infinite family of Siegel zeros implies $\liminf_{n\to\infty}(\gamma_{n+1}-\gamma_n)\log(\gamma_n)/2\pi< 0.4733$ on RH. This refutes the existence of certain strong alternative hypotheses under these assumptions. Our arguments incorporate long Dirichlet polynomials of length $T^{17/14-\varepsilon}$ into the Montgomery--Odlyzko method.

math.NT↗

arXiv2026

On Grünbaum's problem for symmetric configurations

Let $g_n$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as a finite-alphabet squared-error rate-distortion supremum $α_0$. Specialized to the two-point case, i.e., for subsets of Boolean cubes, this gives the explicit lower bound \[g_n\ge (1.160235457\ldots-o(1))^n,\] improving the previous best bound $(2/\sqrt3-o(1))^n$. Using Fix's Gaussian characterization of the rate-distortion problem, we give a numerical three-point construction with exponent base greater than $1.160497831$. Finally, we show that $α_0$ is not attained by any finitely supported distribution.

math.MG↗

arXiv2026

A construction of spherical $5$-designs with $O(d^2)$ points

For every $d\geq1$ we give an explicit equal-weight spherical $5$-design in $\mathbb{S}^{d-1}\subset\mathbb{R}^d$ with at most $72d^2$ points. Our approach utilizes recent construction of complex projective $2$-designs based on Sidon sets.

math.CO↗

arXiv2026

The Hörmander--Bernhardsson extremal function

We characterize the function $φ$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $π$ for which $f(0)=1$. This function, studied by Hörmander and Bernhardsson in 1993, has only real zeros $\pm τ_n$, $n=1,2, \ldots$. Starting from the fact that $n+\frac12-τ_n$ is an $\ell^2$ sequence, established in an earlier paper of ours, we identify $φ$ in the following way. We factor $φ(z)$ as $Φ(z)Φ(-z)$, where $Φ(z)= \prod_{n=1}^\infty(1+(-1)^n\frac{z}{τ_n})$ and show that $Φ$ satisfies a certain second order linear differential equation along with a functional equation either of which characterizes $Φ$. We use these facts to establish an odd power series expansion of $n+\frac12-τ_n$ in terms of $(n+\frac12)^{-1}$ and a power series expansion of the Fourier transform of $φ$, as suggested by the numerical work of Hörmander and Bernhardsson. The dual characterization of $Φ$ arises from a commutation relation that holds more generally for a two-parameter family of differential operators, a fact that is used to perform high precision numerical computations.

math.CA↗

arXiv2025

The basis functions of Fourier interpolation

The basis functions of the Fourier interpolation formula of Radchenko and Viazovska, constructed by means of weakly holomorphic modular forms for the Hecke theta group, are entire functions of order $2$ having interesting time-frequency properties. We give precise size estimates and study the distribution of zeros of these functions. We give in particular asymptotic estimates for the location and the number of extraneous zeros on or close to the real line. This result reveals the surprising existence of Fourier nonuniqueness pairs whose apparent ``excess'' compared to the Fourier uniqueness pair of Radchenko and Viazovska may be made arbitrarily large. Our estimates also show that the basis functions fail to yield a Riesz basis in the Hilbert space used by Kulikov, Nazarov, and Sodin in their recent study of Fourier uniqueness pairs. Some numerical data are presented, suggesting additional fine scale properties.

math.NT↗

arXiv2025

Small Volume Bodies of Constant Width with Tetrahedral Symmetries

For every $n\ge 2$, we construct a body $U_n$ of constant width $2$ in $\mathbb{E}^n$ with small volume and symmetries of a regular $n$-simplex. $U_2$ is the Reuleaux triangle. To the best of our knowledge, $U_3$ was not previously constructed, and its volume is smaller than the volume of other three-dimensional bodies of constant width with tetrahedral symmetries. While the volume of $U_3$ is slightly larger than the volume of Meissner's bodies of width $2$, it exceeds the latter by less than $0.137\%$. For all large $n$, the volume of $U_n$ is smaller than the volume of the ball of radius $0.891$.

math.MG↗

arXiv2025

The Hörmander--Bernhardsson extremal function: A preliminary study

We study the function $φ_1$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $π$ for which $f(0)=1$. This function, first studied by Hörmander and Bernhardsson in 1993, has only real zeros $\pm τ_n$, $n=1,2, \ldots$, and the sequence $(τ_n-n-\frac12)$ has $\ell^2$ norm bounded by $0.13$. The zeros $τ_n$ can be computed by means of a fixed point iteration.

math.FA↗

arXiv2025

On asymptotic Lebesgue's universal covering problem

Universal cover in $\mathbb{E}^{n}$ is a measurable set that contains a congruent copy of any set of diameter 1. Lebesgue's universal covering problem, posed in 1914, asks for the convex set of smallest area that serves as a universal cover in the plane ($n=2$). A simple universal cover in $\mathbb{E}^n$ is provided by the classical theorem of Jung, which states that any set of diameter 1 in an $n$-dimensional Euclidean space is contained in a ball $J_n$ of radius $\sqrt{\tfrac{n}{2n+2}}$; in other words, $J_n$ is a universal cover in $\mathbb{E}^n$. We show that in high dimensions, Jung's ball $J_n$ is asymptotically optimal with respect to the volume, namely, for any universal cover $U \subset \mathbb{E}^n$, $$ {\rm Vol}(U) \ge (1-o(1))^n{\rm Vol}(J_n). $$

math.MG↗

arXiv2025

Small volume bodies of constant width

For every large enough $n$, we explicitly construct a body of constant width $2$ that has volume less than $0.9^n \text{Vol}(\mathbb{B}^{n}$), where $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$. This answers a question of O.~Schramm.

math.MG↗

arXiv2024

On a Gallai-type problem and illumination of spiky balls and cap bodies

We show that any finite family of pairwise intersecting balls in $\mathbb{E}^n$ can be pierced by $(\sqrt{3/2}+o(1))^n$ points improving the previously known estimate of $(2+o(1))^n$. As a corollary, this implies that any $2$-illuminable spiky ball in $\mathbb{E}^n$ can be illuminated by $(\sqrt{3/2}+o(1))^n$ directions. For the illumination number of convex spiky balls, i.e., cap bodies, we show an upper bound in terms of the sizes of certain related spherical codes and coverings. For large dimensions, this results in an upper bound of $1.19851^n$, which can be compared with the previous $(\sqrt{2}+o(1))^n$ established only for the centrally symmetric cap bodies. We also prove the lower bounds of $(\tfrac{2}{\sqrt{3}}-o(1))^n$ for the three problems above.

math.MG↗

arXiv2024

Convex bodies of constant width with exponential illumination number

We show that there exist convex bodies of constant width in $\mathbb{E}^n$ with illumination number at least $(\cos(π/14)+o(1))^{-n}$, answering a question by G. Kalai. Furthermore, we prove the existence of finite sets of diameter $1$ in $\mathbb{E}^n$ which cannot be covered by $(2/\sqrt{3}+o(1))^{n}$ balls of diameter $1$, improving a result by J. Bourgain and J. Lindenstrauss.

math.MG↗

arXiv2024

On Hadwiger's covering problem in small dimensions

Let $H_n$ be the minimal number such that any $n$-dimensional convex body can be covered by $H_n$ translates of interior of that body. Similarly $H_n^s$ is the corresponding quantity for symmetric bodies. It is possible to define $H_n$ and $H_n^s$ in terms of illumination of the boundary of the body using external light sources, and the famous Hadwiger's covering conjecture (illumination conjecture) states that $H_n=H_{n}^s=2^n$. In this note we obtain new upper bounds on $H_n$ and $H_{n}^s$ for small dimensions $n$. Our main idea is to cover the body by translates of John's ellipsoid (the inscribed ellipsoid of the largest volume). Using specific lattice coverings, estimates of quermassintegrals for convex bodies in John's position, and calculations of mean widths of regular simplexes, we prove the following new upper bounds on $H_n$ and $H_n^s$: $H_5\le 933$, $H_6\le 6137$, $H_7\le 41377$, $H_8\le 284096$, $H_4^s\le 72$, $H_5^s\le 305$, and $H_6^s\le 1292$. For larger $n$, we describe how the general asymptotic bounds $H_n\le \binom{2n}{n}n(\ln n+\ln\ln n+5)$ and $H_n^s\le 2^n n(\ln n+\ln\ln n+5)$ due to Rogers and Shephard can be improved for specific values of $n$.

math.MG↗

arXiv2023

A dichotomy for extreme values of zeta and Dirichlet L-functions

We exhibit large values of the Dedekind zeta function of a cyclotomic field on the critical line. This implies a dichotomy whereby one either has improved lower bounds for the maximum of the Riemann zeta function, or large values of Dirichlet $L$-functions on the level of the Bondarenko--Seip bound.

math.NT↗

arXiv2022

Fourier interpolation with zeros of zeta and $L$-functions

We construct a large family of Fourier interpolation bases for functions analytic in a strip symmetric about the real line. Interesting examples involve the nontrivial zeros of the Riemann zeta function and other $L$-functions. We establish a duality principle for Fourier interpolation bases in terms of certain kernels of general Dirichlet series with variable coefficients. Such kernels admit meromorphic continuation, with poles at a sequence dual to the sequence of frequencies of the Dirichlet series, and they satisfy a functional equation. Our construction of concrete bases relies on a strengthening of Knopp's abundance principle for Dirichlet series with functional equations and a careful analysis of the associated Dirichlet series kernel, with coefficients arising from certain modular integrals for the theta group.

math.NT↗

arXiv2019

Linear space properties of $H^p$ spaces of Dirichlet series

We study $H^p$ spaces of Dirichlet series, called $\mathcal{H}^p$, for the range $0<p< \infty$. We begin by showing that two natural ways to define $\mathcal{H}^p$ coincide. We then proceed to study some linear space properties of $\mathcal{H}^p$. More specifically, we study linear functionals generated by fractional primitives of the Riemann zeta function; our estimates rely on certain Hardy--Littlewood inequalities and display an interesting phenomenon, called contractive symmetry between $\mathcal{H}^p$ and $\mathcal{H}^{4/p}$, contrasting the usual $L^p$ duality. We next deduce general coefficient estimates, based on an interplay between the multiplicative structure of $\mathcal{H}^p$ and certain new one variable bounds. Finally, we deduce general estimates for the norm of the partial sum operator $\sum_{n=1}^\infty a_n n^{-s}\mapsto \sum_{n=1}^N a_n n^{-s}$ on $\mathcal{H}^p$ with $0< p \le 1$, supplementing a classical result of Helson for the range $1<p<\infty$. The results for the coefficient estimates and for the partial sum operator exhibit the traditional schism between the ranges $1\le p \le \infty$ and $0<p<1$.

math.FA↗

arXiv2018

On certain sums over ordinates of zeta-zeros II

Let $γ$ denote the imaginary parts of complex zeros $ρ= β+iγ$ of $ζ(s)$. The problem of analytic continuation of the function $G(s) := \sum\limits_{γ> 0}γ^{-s}$ to the left of the line $\Re s = -1$ is investigated, and its Laurent expansion at the pole $s=1$ is obtained. Estimates for the second moment on the critical line $\int_1^T|G(1/2+it)|^2\,dt$ are revisited. This paper is a continuation of work begun by the second author in 2001.

math.NT↗

arXiv2018

Pseudomoments of the Riemann zeta function

The $2$kth pseudomoments of the Riemann zeta function $ζ(s)$ are, following Conrey and Gamburd, the $2k$th integral moments of the partial sums of $ζ(s)$ on the critical line. For fixed $k>1/2$, these moments are known to grow like $(\log N)^{k^2}$, where $N$ is the length of the partial sum, but the true order of magnitude remains unknown when $k\le 1/2$. We deduce new Hardy--Littlewood inequalities and apply one of them to improve on an earlier asymptotic estimate when $k\to\infty$. In the case $k<1/2$, we consider pseudomoments of $ζ^α(s)$ for $α>1$ and the question of whether the lower bound $(\log N)^{k^2α^2}$ known from earlier work yields the true growth rate. Using ideas from recent work of Harper, Nikeghbali, and Radziwi{łł} and some probabilistic estimates of Harper, we obtain the somewhat unexpected result that these pseudomements are bounded below by $\log N$ to a power larger than $k^2α^2$ when $k<1/e$ and $N$ is sufficiently large.

math.FA↗