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Norbert Hegyvári

Publications and source records attributed to Norbert Hegyvári.

At least 19 recordsLinked to original sources
arXiv2026

Arithmetic Progressions in a Random Binary Subset-Sum Set

Let $u=(u_n)_{n\ge0}$ be a binary sequence, and define \[ X_0=1,\qquad X_{n+1}=2X_n+u_n \qquad(n\ge0). \] Let $A_u$ be the set of all nonempty finite subset sums of the sequence $(X_n)$, and let $L_u(N)$ denote the maximum length of an arithmetic progression contained in $A_u\cap[1,N]$. We prove that there are absolute constants $c>0$ and $N_0\geq 1$ such that, for every binary sequence $u$, \[ L_u(N)\ge \exp\!\left(c\sqrt{\frac{\log N}{\log\log N}}\right) \] for all $N\geq N_0$. Moreover, if the random variables $u_n$ are independent and uniformly distributed on $\{0,1\}$, then, almost surely, \[ L_u(N)\ll_u N^{2/3}\exp\!\left(C\sqrt{\log N\log\log N}\right), \] where $C>0$ is an absolute constant. Furthermore, every eventually periodic binary sequence satisfies $L_u(N)\gg_u N^{1/2}$.

math.CO↗
arXiv2026

Subset-Sum Density Realization in Locally Finite Abelian Groups

Let $G$ be a countable locally finite abelian group, and let \[ G_1\leq G_2\leq\cdots, \qquad \bigcup_{i\geq1}G_i=G, \] be any filtration of $G$ by finite subgroups. For $A\subseteq G$, let $\mathcal P(A)$ denote the set of all finite subset sums of elements of $A$, and let $2G:=\{2g:g\in G\}$. We prove that $|2G|=\infty$ if and only if for every filtration and every interval $[α,β]\subseteq[0,1]$, there exists $A\subseteq G$ such that the set of limit points of \[ \left( \frac{|\mathcal P(A)\cap G_i|}{|G_i|} \right)_{i\geq1} \] is exactly $[α,β]$. We also prove a positive-density result that does not require $|2G|=\infty$: if $G$ is infinite and $\mathcal P(A)$ has positive upper density along the given filtration, then there is an infinite set $B\subseteq \mathcal P(A)$ such that $B+B\subseteq \mathcal P(A)$. Combining this with the realization theorem, we show that, when $|2G|=\infty$, every interval $[α,β]$ with $0\leqα\leqβ\leq1$ and $β>0$ can be realized by a set $A$ with this additional property.

math.CO↗
arXiv2026

Packing sets under finite groups via algebraic incidence structures

Let $G$ be a finite group acting on a vector space $V = \mathbb{F}_p^n$ over a prime field. Given finite sets $S \subset G$ and $E \subset V$, we study the restricted orbit union $S(E) = \bigcup_{g\in S} g(E)$ and establish quantitative lower bounds for $|S(E)|$ in terms of $|S|$, $|E|$, and natural structural conditions. This finite field packing problem has connections to distance geometry, configuration counting, and expanding graphs. For $G = SL_2(\mathbb{F}_p)$ acting on $\mathbb{F}_p^2$, we prove that $$|S(E)| \gg \min\left\lbrace p^2, \frac{|S||E|}{p^2}\right\rbrace,$$ which is sharp. Under geometric non-concentration conditions on $E$ and subgroup-avoidance hypotheses on $S$, we obtain a power-saving improvement of the form $$|S(E)|\gg \min \left\lbrace p^2, ~\max\left\lbrace\frac{|S||E|}{pk}, ~\frac{|S|^{\frac{1}{2}}|E|}{p^{\frac{1-ε}{2}}k^{\frac{1}{2}}}\right\rbrace \right\rbrace,$$ where $k$ bounds the radial multiplicity of $E$. For small sets $|E| \leq p$, we establish optimal bounds using weighted incidence theory. Analogous results are proved for the first Heisenberg group $\mathbb{H}_1(\mathbb{F}_p)$ acting on $\mathbb{F}_p^3$. Our approach reformulates the problem as an incidence question in a bipartite action graph. The proofs combine Fourier analytic techniques, energy estimates, point-line incidence bounds, and area-energy inequalities for skew dot products. The methods extend classical sum-product type problems and incidence theory to noncommutative group actions.

math.CO↗
arXiv2026

On the distribution of additive energy revisited

This paper extends the investigation of energy distribution in finite settings, which is related to the results established in [H]. We analyze the distribution of multiplicative energies using Fourier analytical methods and random structures. Our results provide new structural insights into energy phenomena in finite fields, complementing the earlier discrete analysis. Additionally, we provide an estimate for the smallest $k$ such that the $k$-fold product set $A^k$ covers the entire field $\mathbb{F}$, given that $A$ has small doubling.

math.CO↗
arXiv2025

Note on the sumset of squares

It is proved that for any non-empty finite subset $Q$ of the square numbers, $ |Q+Q|\geq C'|Q|(\log |Q|)^{1/3+o(1)} $. This result essentially is proved -- with the same tools -- by Mei-Chu Chang. See in J. Funct. Anal. 207 (2004), no 2, 444-460. So the author will withdraw this ArXiv file.

math.CO↗
arXiv2025

An elementary question of Erdős and Graham

Let $A_k=\{r(k-r): 1\leq r \leq k-1\}$. Erd\H os and Graham asked about the cardinality of the set of common elements. We answer this elementary question and apply our result to a sum-product type result.

math.HO↗
arXiv2024

On the structures of subset sums in higher dimension

A given subset $A$ of natural numbers is said to be complete if every element of $\N$ is the sum of distinct terms taken from $A$. This topic is strongly connected to the knapsack problem which is known to be NP complete. The main goal of the paper is to study the structure of subset sums in a higher dimension. We show 'dense' sets and generalized arithmetic progrssions in subset sums of certain sets.

math.CO↗
arXiv2023

Covering shrinking polynomials by quasi progressions

Erd\H os introduced the quantity $S=T\sum^T_{i=1}X_i$, where $X_1,\dots, X_T$ are arithmetic progressions, and cover the square numbers up to $N$. He conjectured that $S$ is close to $N$, i.e. the square numbers cannot be covered "economically" by arithmetic progressions. Sárközy confirmed this conjecture and proved that $S\geq cN/\log^2N$. In this paper, we extend this to shrinking polynomials and so-called $\{X_i\}$ quasi progressions.

math.CO↗
arXiv2023

Estimation of function's supports under arithmetic constraints

The well-known $|supp(f)||supp(\widehat{f}|\geq |G|$ inequality gives lower estimation of each supports. In the present paper we give upper estimation under arithmetic constrains. The main notion will be the additive energy which plays a central role in additive combinatorics. We prove an uncertainty inequality that shows a trade-off between the total changes of the indicator function of a subset $A\subseteq \mathbb F^n_2$ and the additive energy of $A$ and the Fourier spectrum.

math.CO↗
arXiv2023

On arithmetic sums of Cantor-type sequences of integers

We are looking for integer sets that resemble classical Cantor set and investigate the structure of their sum sets. Especially we investigate $FS(B)$ the subset sum of sequence type $B=\{\lfloor p^nα\rfloor\}^\infty_{n=0}$. When $p=2$, then we prove $FS(B)+FS(B)=\N$ by analogy with the Cantor set, and some structure theorem for $p>2$

math.CO↗
arXiv2023

On the distribution of subset sums of certain sets in $\mathbb{Z}^2_p$

A given subset $A$ of natural numbers is said to be complete if every element of $\mathbb{N}$ is the sum of distinct terms taken from $A$. This topic is strongly connected to the knapsack problem which is known to be NP complete. Interestingly if $A$ and $B$ are complete sequences then $A\times B$ is not necessarily complete in $\mathbb{N}^2$. In this paper we consider a modular version of this problem, motivated by the communication complexity problem of [2].

math.CO↗
arXiv2022

Some remarks on the distribution of additive energy

The aim of this note is two-fold. In the first part of the paper we are going to investigate an inverse problem related to additive energy. In the second, we investigate how dense a subset of a finite structure can be for a given additive energy.

math.CO↗
arXiv2021

On a Communication Complexity problem in Combinatorial Number Theory

The original knapsack problem is well known to be NP-complete. In a multidimensional version one have to decide whether a $p\in \N^k$ is in a sumset-sum of a set $X \subseteq \N^k$ or not. In this paper we are going to investigate a communication complexity problem related to this.

cs.DM↗
arXiv2013

Conditional expanding bounds for two-variables functions over prime fields

In this paper we provide in $\bFp$ expanding lower bounds for two variables functions $f(x,y)$ in connection with the product set or the sumset. The sum-product problem has been hugely studied in the recent past. A typical result in $\bFp^*$ is the existenceness of $Δ(α)>0$ such that if $|A|\asymp p^α$ then $$ \max(|A+A|,|A\cdot A|)\gg |A|^{1+Δ(α)}, $$ Our aim is to obtain analogous results for related pairs of two-variable functions $f(x,y)$ and $g(x,y)$: if $|A|\asymp|B|\asymp p^α$ then $$ \max(|f(A,B)|,|g(A,B)|)\gg |A|^{1+Δ(α)} $$ for some $Δ(α)>0$.

math.NT↗
arXiv2013

A Structure result for bricks in Heisenberg groups

We show that for a sufficiently big \textit{brick} $B$ of the $(2n+1)$-dimensional Heisenberg group $H_n$ over the finite field $\mathbb{F}_p$, the product set $B\cdot B$ contains at least $|B|/p$ many cosets of some non trivial subgroup of $H_n$.

math.NT↗
arXiv2012

Distribution of residues in approximate subgroups of $\mathbb{F}_p^*$

We extend a result due to Bourgain on the uniform distribution of residues by proving that subsets of the type $f(I)\cdot H$ is equidistributed (as $p$ tends to infinity) where $f$ is a polynomial, $I$ is an interval of $\Fp$ and $H$ is an approximate subgroup of $\mathbb{F}_p^*$ with size larger than polylogarithmic in $p$.

math.NT↗
arXiv2012

A note on Freiman models in Heisenberg groups

Green and Ruzsa recently proved that for any $s\ge2$, any small squaring set $A$ in a (multiplicative) abelian group, i.e. $|A\cdot A|<K|A|$, has a Freiman $s$-model: it means that there exists a group $G$ and a Freiman $s$-isomorphism from $A$ into $G$ such that $|G|<f(s,K)|A|$. In an unpublished note, Green proved that such a result does not necessarily hold in non abelian groups if $s\ge64$. The aim of this paper is improve Green's result by showing that it remains true under the weaker assumption $s\ge6$.

math.NT↗