arXiv subjects

Szilárd Gy. Révész

Publications and source records attributed to Szilárd Gy. Révész.

At least 19 recordsLinked to original sources

arXiv2026

On a question of Gowers related to Littlewood's conjecture

In a blogpost in 2009, Gowers raised a possible approach to Littlewood's conjecture in Diophantine approximation, leading to a question about the existence of sufficiently many points in the unit cube such that the ``hyperbolic distance" between any two of them is large. In this note we answer this question by an explicit construction. This shows that this approach to prove Littlewood's conjecture cannot work, unless some further refinements are added.

math.NT↗

arXiv2026

Duality for Delsarte's extremal problem on compact Gelfand pairs

We study Delsarte-type problems for positive definite functions on compact Gelfand pairs as infinite-dimensional linear programming problems. This setup includes, as a particular case, the case of compact Abelian groups. Depending on the restriction on the signs of the functions, we obtain two important particular cases, the Turán and Delsarte problems. These problems have been studied in relation to number theory, sphere packing, and statistics. In this paper, we describe their duals and prove a strong duality statement.

math.CA↗

arXiv2026

Conjectures of Bernstein and Erd\H os for weighted Lagrange interpolation on the halfline with exponential weights

Let I=[a,b] and consider the degree n Lagrange interpolation at the nodes x, where x\in S:={x=(x_0,x_1,...,x_n):a=x_0 m_j(x). The conjectures were proved by Kilgore and de Boor--Pinkus in 1978. Since then, analogous results were obtained only for a few cases when interpolation is made to certain very special spaces of polynomials, or when we apply weighted interpolation with rather special weights. Worse than that, it turned out that published proofs of results on infinite intervals and weighted interpolation were seriously flawed. Here we prove the Bernstein and Erd\H os Conjectures for the case of exponentially weighted polynomials on the halfline. This is the first proof of these conjectures in a situation where, contrary to all existing successful proofs, we encounter singularity of certain derivative matrices.

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arXiv2026

Turán type oscillation inequalities in $L^q$ norm on the boundary of convex polygonal domains

In 1939 Pál Turán and János Erőd initiated the study of lower estimations of maximum norms of derivatives of polynomials, in terms of the maximum norms of the polynomials themselves, on convex domains of the complex plane. As a matter of normalization they considered the family $\mathcal{P}_n(K)$ of degree $n$ polynomials with all zeros lying in the given convex, compact subset $K\Subset {\mathbb C}$. While Turán obtained the first results for the interval $I:=[-1,1]$ and the disk $D:=\{ z\in {\mathbb C}~:~ |z|\le 1\}$, Erőd extended investigations to other compact convex domains, too. The order of the optimal constant was found to be $\sqrt{n}$ for $I$ and $n$ for $D$. It took until 2006 to clarify that all compact convex \emph{domains} (with nonempty interior), follow the pattern of the disk, and admit an order $n$ inequality. For $L^q(\partial K)$ norms with any $1\le q <\infty$ we obtained order $n$ results for various classes of domains. Further, in the generality of all convex, compact domains we could show a $c n/\log n$ lower bound together with an $O(n)$ upper bound for the optimal constant. Also, we conjectured that all compact convex domains admit an order $n$ Turán type inequality. Here we prove this for all \emph{polygonal} convex domains and any $0< q <\infty$.

math.CV↗

arXiv2026

Duality for Delsarte's extremal problem on locally compact Abelian groups

The Delsarte extremal problem for positive definite functions, originally introduced by Delsarte in coding theory to bound the size of error-correcting codes, has since found applications in diverse areas such as sphere packing, Fuglede's spectral set conjecture, and $1$-avoiding sets. Recent developments have established the existence of extremizers in fairly general settings and identified precise linear programming dual formulations, together with strong duality results, in several important cases including finite groups and $\mathbb{R}^d$. In this paper, we consider a generalized Delsarte problem on locally compact Abelian groups, providing a natural framework for harmonic analysis. We extend both the normalization and the objective functional to encompass a wide range of previously studied cases, while avoiding restrictive topological assumptions common in the literature. Within this general setting, we derive the corresponding dual problem and prove a strong duality theorem, thereby unifying and extending earlier results. Naturally, our proof uses harmonic analysis, but the key is a functional analytic approach which distinguishes our proof from existing methods.

math.FA↗

arXiv2024

On extremal problems of Delsarte type for positive definite functions on LCA groups

A unifying framework for some extremal problems on locally compact Abelian groups is considered, special cases of which include the Delsarte and Turán extremal problems. A slight variation of the extremal problem is introduced and the different formulations are studied for equivalence. Extending previous work, a general result on existence of extremal functions for the new variant is proved under a certain general topological condition.

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arXiv2024

The Carlson-type zero-density theorem for the Beurling zeta function

In a previous paper we proved a Carlson type density theorem for zeroes in the critical strip for Beurling zeta functions satisfying Axiom A of Knopfmacher. There we needed to invoke two additonal conditions, the integrality of the norm (Condition B) and an "average Ramanujan Condition" for the arithmetical function counting the number of different Beurling integers of the same norm (Condition G). Here we implement a new approach of Pintz, using the classic zero detecting sums coupled with Halász' method, but otherwise arguing in an elementary way avoiding e.g. large sieve type inequalities. This way we give a new proof of a Carlson type density estimate--with explicit constants--avoiding any use of the two additional conditions, needed earlier. Therefore it is seen that the validity of a Carlson-type density estimate does not depend on any extra assumption--neither on the functional equation, present for the Selberg class, nor on growth estimates of coefficients say of the "average Ramanujan type"-but is a general property, presenting itself whenever the analytic continuation is guaranteed by Axiom A.

math.NT↗

arXiv2024

New zero-density estimates for the Beurling $ζ$ function

In two previous papers the second author proved some Carlson type density theorems for zeroes in the critical strip for Beurling zeta functions satisfying Axiom A of Knopfmacher. In the first of these invoking two additonal conditions were needed, while in the second an explicit, fully general result was obtained. Subsequently, Frederik Broucke and Gregory Debruyne obtained, via a different method, a general Carlson type density theorem with an even better exponent, and recently Frederik Broucke improved this further, getting $N(σ,T) \le T^{a(1-σ)}$ with any $a>\dfrac{4}{1-θ}$. Broucke employed a new mean value estimate of the Beurling zeta function, while he did not use the method of Halász and Montgomery. Here we elaborate a new approach of the first author, using the classical zero detecting sums coupled with a kernel function technique and Halász' method, but otherwise arguing in an elementary way avoiding e.g. mean value estimates for Dirichlet polynomials. We will make essential use of the additional assumptions that the Beurling system of integers consists of natural numbers, and that the system satisfies the Ramanujan condition, too. This way we give a new variant of the Carlson type density estimate with similar strength as Turán's 1954 result for the Riemann $ζ$ function, coming close even to the Density Hypothesis for $σ$ close to 1.

math.NT↗

arXiv2023

Kahane's upper density and syndetic sets in LCA groups

Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line. Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group $G$, a set $S\subset G$ is syndetic, if there exists a compact subset $C\Subset G$ such that $SC=G$. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of Fürstenberg says that once a subset $A \subset {\mathbb Z}$ has positive a.u.u.d., then its difference set $A-A$ is syndetic. The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000's several constructions were worked out to generalize it from the base cases of ${\mathbb Z}^d$ and ${\mathbb R}^d$. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups. Here we work out various versions in a general LCA group $G$ of the classical statement that if a set $S\subset G$ has positive asymptotic uniform upper density, then the difference set $S-S$ is syndetic.

math.CA↗

arXiv2023

On the weighted trigonometric Bojanov-Chebyshev extremal problem

We investigate the weighted Bojanov-Chebyshev extremal problem for trigonometric polynomials, that is, the minimax problem of minimizing $\|T\|_{w,C({\mathbb T})}$, where $w$ is a sufficiently nonvanishing, upper bounded, nonnegative weight function, the norm is the corresponding weighted maximum norm on the torus ${\mathbb T}$, and $T$ is a trigonometric polynomial with prescribed multiplicities $ν_1,\ldots,ν_n$ of root factors $|\sin(π(t-z_j))|^{ν_j}$. If the $ν_j$ are natural numbers and their sum is even, then $T$ is indeed a trigonometric polynomial and the case when all the $ν_j$ are 1 covers the Chebyshev extremal problem. Our result will be more general, allowing, in particular, so-called generalized trigonometric polynomials. To reach our goal, we invoke Fenton's sum of translates method. However, altering from the earlier described cases without weight or on the interval, here we find different situations, and can state less about the solutions.

math.CA↗

arXiv2023

Fenton type minimax problems for sum of translates functions

Following P. Fenton, we investigate sum of translates functions $F(\mathbf{x},t):=J(t)+\sum_{j=1}^n ν_j K(t-x_j)$, where $J:[0,1]\to {\underline{\mathbb{R}}}:=\mathbb{R}\cup\{-\infty\}$ is a "sufficiently non-degenerate" and upper-bounded "field function", and $K:[-1,1]\to {\underline{\mathbb{R}}}$ is a fixed "kernel function", concave both on $(-1,0)$ and $(0,1)$, $\mathbf{x}:=(x_1,\ldots,x_n)$ with $0\le x_1\le\dots\le x_n\le 1$, and $ν_1,\dots,ν_n>0$ are fixed. We analyze the behavior of the local maxima vector $\mathbf{m}:=(m_0,m_1,\ldots,m_n)$, where $m_j:=m_j(\mathbf{x}):=\sup_{x_j\le t\le x_{j+1}} F(\mathbf{x},t)$, with $x_0:=0$, $x_{n+1}:=1$; and study the optimization (minimax and maximin) problems $\inf_{\mathbf{x}}\max_j m_j(\mathbf{x})$ and $\sup_{\mathbf{x}}\min_j m_j(\mathbf{x})$. The main result is the equality of these quantities, and provided $J$ is upper semicontinuous, the existence of extremal configurations and their description as equioscillation points $\mathbf{w}$. In our previous papers we obtained results for the case of singular kernels, i.e., when $K(0)=-\infty$ and the field $J$ was assumed to be upper semicontinuous. In this work we get rid of these assumptions and prove common generalizations of Fenton's and our previous results, arriving at the greatest generality in the setting of concave kernel functions.

math.CA↗

arXiv2023

Intertwining of maxima of sum of translates functions with nonsingular kernels

In previous papers we investigated so-called sum of translates functions $F({\mathbf{x}},t):=J(t)+\sum_{j=1}^n ν_j K(t-x_j)$, where $J:[0,1]\to \underline{\mathbb{R}}:={\mathbb{R}}\cup\{-\infty\}$ is a "sufficiently nondegenerate" and upper-bounded "field function", and $K:[-1,1]\to \underline{\mathbb{R}}$ is a fixed "kernel function", concave both on $(-1,0)$ and $(0,1)$, and also satisfying the singularity condition $K(0)=\lim_{t\to 0} K(t)=-\infty$. For node systems ${\mathbf{x}}:=(x_1,\ldots,x_n)$ with $x_0:=0\le x_1\le\dots\le x_n\le 1=:x_{n+1}$, we analyzed the behavior of the local maxima vector ${\mathbf{m}}:=(m_0,m_1,\ldots,m_n)$, where $m_j:=m_j({\mathbf{x}}):=\sup_{x_j\le t\le x_{j+1}} F({\mathbf{x}},t)$. Among other results we proved a strong intertwining property: if the kernels are also decreasing on $(-1,0)$ and increasing on $(0,1)$, and the field function is upper semicontinuous, then for any two different node systems there are $i,j$ such that $m_i({\mathbf{x}}) m_j({\mathbf{y}})$. Here we partially succeed to extend this even to nonsingular kernels.

math.CA↗

arXiv2023

On the weighted Bojanov-Chebyshev Problem and the sum of translates method of Fenton

Minimax and maximin problems are investigated for a special class of functions on the interval $[0,1]$. These functions are sums of translates of positive multiples of one kernel function and a very general external field function. Due to our very general setting the obtained minimax, equioscillation, and characterization results extend those of Bojanov, Fenton, Hardin, Kendall, Saff and Ambrus, Ball, Erdélyi. Moreover, we discover a surprising intertwining phenomenon of interval maxima, which provides new information even in the most classical extremal problem of Chebyshev.

math.CA↗

arXiv2023

The method of Pintz for the Ingham question about the connection of distribution of $ζ$-zeros and order of the error in the PNT in the Beurling context

We prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. These are concerned with the question of Ingham who asked for optimal and explicit order estimates for the error term $Δ(x):=ψ(x)-x$, given any zero-free region $D(η):=\{s=σ+it\in\mathbb{C}~:~ σ:=\Re s \ge 1-η(t)\}$. In the classical case essentially sharp results are due to some 40 years old work of Pintz. Here we consider a given a system of Beurling primes $\mathcal{P}$, the generated arithmetical semigroup $\mathcal{G}$ and the corresponding integer counting function $N(x)$, and the corresponding error term $Δ_{\mathcal{G}}(x):=ψ_{\mathcal{G}}(x)-x$ in the PNT of Beurling, where $ψ_{\mathcal{G}}(x)$ is the Beurling analog of $ψ(x)$. First we prove that if the Beurling zeta function $ζ_{\mathcal{G}}$ does not vanish in $D(η)$, then the extension of Pintz' result holds: $|Δ_{\mathcal{G}}(x)| \le x\exp(-(1-\varepsilon)ω_η(x))~(x>x_0(\varepsilon))$, where $ω_η(y)$ is the naturally occurring conjugate function to $η(t)$, introduced into the field by Ingham. In the second part we prove a converse: if $ζ_{\mathcal{G}}$ has an infinitude of zeroes in the given domain, then analogously to the classical case, $|Δ_{\mathcal{G}}(x)| \ge x\exp(-(1+\varepsilon)ω_η(x))$ holds "infinitely often". This also shows that both main results are sharp apart from the arbitrarily small $\varepsilon>0$.

math.NT↗

arXiv2023

A homeomorphism theorem for sums of translates

For a fixed positive integer $n$ consider continuous functions $ K_1,\dots$, $ K_n:[-1,1]\to \mathbb{R}\cup\{-\infty\}$ that are concave and real valued on $[-1,0)$ and on $(0,1]$, and satisfy $K_j(0)=-\infty$. Moreover, let $J:[0,1]\to \mathbb{R}\cup\{-\infty\}$ be upper bounded and such that $[0,1]\setminus J^{-1}(\{-\infty\})$ has at least $n+1$ elements, but it is arbitrary otherwise. For $x_0:=0<x_1<\dots< x_n \le x_{n+1}:=1$, so called nodes, and for $t\in [0,1]$ consider the sum of translates function $F(x_1,\ldots,x_n,t):=J(t)+\sum_{j=1}^n K_j(t-x_j)$, and the vector of interval maximum values $m_j:=m_j(x_1,\ldots,x_n):=\max_{t\in [x_j,x_{j+1}]}F(x_1,\ldots,x_n,t)$ ($j=0,1,\ldots,n$). We describe the structure of the arising interval maxima as the nodes run over the $n$-dimensional simplex. Applications presented here range from abstract moving node Hermite-Fejér interpolation for generalized algebraic and trigonometric polynomials via Bojanov's problem to more abstract results of interpolation theoretic flavour.

math.CA↗

arXiv2022

Oscillation of the remainder term in the prime number theorem of Beurling, "caused by a given zeta-zero"

Continuing previous study of the Beurling zeta function, here we prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. First, we address the question of Littlewood, who asked for explicit oscillation results provided a zeta-zero is known. We prove that given a zero $ρ_0$ of the Beurling zeta function $ζ_P$ for a given number system generated by the primes $P$, the corresponding error term $Δ_P(x):=ψ_{P}(x)-x$, where $ψ_{P}(x)$ is the von Mangoldt summatory function shows oscillation in any large enough interval, as large as $(π/2-\varepsilon) x^{\Re ρ_0}/|ρ_0|$. The somewhat mysterious appearance of the constant $π/2$ is explained in the study. Finally, we prove as the next main result of the paper the following: given $\varepsilon>0$, there exists a Beurling number system with primes $P$, such that $|Δ_P(x)| \le (π/2+\varepsilon)x^{\Re ρ_0}/|ρ_0|$. In this second part a nontrivial construction of a low norm sine polynomial is coupled by the application of the wonderful recent prime random approximation result of Broucke and Vindas, who sharpened the breakthrough probabilistic construction due to Diamond, Montgomery and Vorhauer.

math.NT↗

arXiv2022

Distribution of Beurling primes and zeroes of the Beurling zeta function I. Distribution of the zeroes of the zeta function of Beurling

We prove three results on the density resp. local density and clustering of zeros of the Beurling zeta function $ζ(s)$ close to the one-line $σ:=\Re s=1$. The analysis here brings about some news, sometimes even for the classical case of the Riemann zeta function. Theorem 4 provides a zero density estimate, which is a complement to known results for the Selberg class. Note that density results for the Selberg class rely on use of the functional equation of $ζ$, which we do not assume in the Beurling context. In Theorem 5 we deduce a variant of a well-known theorem of Turán, extending its range of validity even for rectangles of height only $h=2$. In Theorem 6 we will extend a zero clustering result of Ramachandra from the Riemann zeta case. A weaker result -- which, on the other hand, is a strong sharpening of the average result from the classic book \cite{Mont} of Montgomery -- was worked out by Diamond, Montgomery and Vorhauer. Here we show that the obscure technicalities of the Ramachandra paper (like a polynomial with coefficients like $10^8$) can be gotten rid of, providing a more transparent proof of the validity of this clustering phenomenon.

math.NT↗

arXiv2022

Delsarte's Extremal Problem and Packing on Locally Compact Abelian Groups

Let G be a locally compact Abelian group, and let X, Y be two open sets in G. We investigate the extremal constant C(X,Y) defined to be the supremum of integrals of functions f from the class F(X,Y), where F(X,Y) is the family of positive definite functions f on G such that f(0) = 1, the positive part of f is supported in X, and its negative part is supported in Y. In the case when X=Y, the problem is exactly the so-called Turán problem for the set X. When Y= G, i.e., there is a restriction only on the set of positivity of f, we obtain the Delsarte problem. The Delsarte problem is the sharpest Fourier analytic tool to study packing density by translates of a given "master copy" set, which was studied first in connection with packing densities of Euclidean balls. We give an upper estimate of the constant C(X,Y) in the situation when the set X satisfies a certain packing type condition. This estimate is given in terms of the asymptotic uniform upper density of sets in locally compact Abelian groups.

math.CA↗