Gábor Somlai

activity

20092026

most citedOn the discrete Fuglede and Pompeiu problems

22 citations · 54 across the 25 of their papers we have counts for

collaborators

32 papers

math.GR2026

Tilings of symmetric and alternating groups by conjugacy classes

Gábor Somlai, Binzhou Xia, Sanming Zhou

We study tilings of symmetric and alternating groups by conjugacy classes. For $S_{n}$ , the case where the conjugacy class consists of transpositions was first investigated by Rothau…

math.NT2026

Non-existence of sets with few special directions

Luca Ghidelli, Gergely Kiss, Gábor Somlai

Let $p$ be an odd prime and let $S \subseteq F_{p}^{2}$ have cardinality divisible by $p$ . We prove that, for all sufficiently large primes $p$ , no such set has exactly four spe…

math.CO2026

Fuglede's Conjecture on Cyclic Groups of Square-Free Order: The Case of Rapidly Growing Prime Factors

Gábor Somlai

The main result of the paper is that an inductive argument is established to prove the Fuglede's conjecture for an infinite sequence of square-free order cyclic groups. The tile-to…

math.GR2025

Dihedral groups of square-free order are DCI-groups

István Kovács, Gábor Somlai

A finite group $G$ is a called a DCI-group if any two isomorphic Cayley digraphs of $G$ are also isomorphic via an automorphism of $G$ . If $G$ is a non-abelian generalised dihedral…

math.NT2025

Lower bounds for mask polynomials with many cyclotomic divisors

Gergely Kiss, Izabella Łaba, Caleb Marshall +1

Given a nonempty set $A \subset N \cup {0}$ , define the mask polynomial $A (X) = \sum_{a \in A} X^{a}$ . Suppose that there are $s_1,\dots,s_k\in\nn\setminus\{1\}$ such that the…

math.CO2024

Functional tilings and the Coven-Meyerowitz tiling conditions

Gergely Kiss, Itay Londner, Máté Matolcsi +1

Coven and Meyerowitz formulated two conditions which have since been conjectured to characterize all finite sets that tile the integers by translation. By periodicity, this conject…

math.CA2024★ 1 cited

Spectrality of the product does not imply that the components are spectral

Gábor Somlai

Greenfeld and Lev conjectured that the Cartesian product of two sets $A$ and $B$ is spectral if and only if $A$ and $B$ are spectral. We construct a counterexample to this fact usi…

math.CO2024

A lonely weak tile

Gergely Kiss, Itay Londner, Máté Matolcsi +1

The notion of weak tiling was a key ingredient in the proof of Fuglede's spectral set conjecture for convex bodies \cite{conv}, due to the fact that every spectral set tiles its co…

math.NT2023

On polynomials of small range sum

Gergely Kiss, Ádám Markó, Zoltán Lóránt Nagy +1

In order to reprove an old result of Rédei's on the number of directions determined by a set of cardinality $p$ in $F_{p}^{2}$ , Somlai proved that the non-constant polynomials…

math.CA2023★ 1 cited

Spectral Tile Direction in the Group $Z_{p^{2}} \times Z_{q^{2}} \times Z_{r}$

Thomas Fallon, Gergely Kiss, Azita Mayeli +1

Let $p$ , $q$ , and $r$ be distinct primes such that $p^{2} q^{2} < r$ . We prove that every spectral set in the cyclic group $Z_{p^{2} q^{2} r}$ is a tile. Since the reverse direction is…

math.NT2023

Decompositions of the positive real numbers into disjoint sets closed under addition and multiplication

Gergely Kiss, Gábor Somlai, Tamás Terpai

The main purpose of this paper is to prove that the positive real numbers can be decomposed into finitely many disjoint pieces which are also closed under addition and multiplicati…

math.NT2022

A new proof of Rédei's theorem on the number of directions

Gabor Somlai

Rédei and Megyesi proved that the number of directions determined by a $p$ element subset of $F_{p}^{2}$ is either $1$ or at least $\frac{p + 3}{2}$ . The same result was indepen…

math.CO2022

Tiling and weak tiling in $(Z_{p})^{d}$

Gergely Kiss, Dávid Matolcsi, Máté Matolcsi +1

We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary $p$ -groups $(Z_{p})^{d}$ , we introduce an averaging pro…

math.MG2022

Optimal embedded and enclosing isosceles triangles

Aron Ambrus, Monika Csikos, Gergely Kiss +2

Given a triangle $Δ$ , we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of $Δ$ with respect to area and perimeter. This problem wa…

math.GR2022

On the diameter of Cayley graphs of classical groups with generating sets containing a transvection

Martino Garonzi, Zoltán Halasi, Gábor Somlai

A well-known conjecture of Babai states that if $G$ is any finite simple group and $X$ is a generating set for $G$ , then the diameter of the Cayley graph $C a y (G, X)$ is bounded by $…

math.CO2022

CI-property of $C_{p}^{2} \times C_{n}$ and $C_{p}^{2} \times C_{q}^{2}$ for digraphs

István Kovács, Mikhail Muzychuk, Péter P. Pálfy +2

We prove that the direct product of two coprime order elementary abelian groups of rank two, as well as the direct product of a cyclic group of prime order and a cyclic group of sq…

math.CO2021

Special directions on the finite affine plane

Gergely Kiss, Gábor Somlai

In this paper we study the number of special directions of sets of cardinality divisible by $p$ on a finite plane of characteristic $p$ , where $p$ is a prime. We show that there is…

math.CA2021★ 2 cited

Spectral sets and tiles in $Z_{p}^{2} \times Z_{q}^{2}$

Thomas Fallon, Gergely Kiss, Gábor Somlai

In this paper we study Fuglede's conjecture on the direct product of two cyclic groups. After collecting the known results we prove that Fuglede's conjecture holds on $\mathbb{Z}_{…

math.CA2020★ 20 cited

Fuglede's conjecture holds for cyclic groups of order $pq r s$

Gergely Kiss, Romanos Diogenes Malikiosis, Gábor Somlai +1

The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of square-free order, initiated by Laba and Meyerowitz, but the spectral-tile directi…

math.MG2020

Minimum area isosceles containers

Gergely Kiss, János Pach, Gábor Somlai

We show that every minimum area isosceles triangle containing a given triangle $T$ shares a side and an angle with $T$ . This proves a conjecture of Nandakumar motivated by a comput…

math.CA2019

Fuglede's conjecture holds on $Z_{p}^{2} \times Z_{q}$

Gergely Kiss, Gábor Somlai

The study of Fuglede's conjecture on the direct product of elementary abelian groups was initiated by Iosevich et al. For the product of two elementary abelian groups the conjectur…

math.GR2019★ 4 cited

The Cayley isomorphism property for $Z_{p}^{3} \times Z_{q}$

Mikhail Muzychuk, Gábor Somlai

For every pair of distinct primes $p$ , $q$ we prove that $Z_{p}^{3} \times Z_{q}$ is a CI-group with respect to binary relational structures.

math.CA2019★ 2 cited

Spectral sets in $Z_{p^{2} q r}$ tile

Gábor Somlai

We prove the every spectral set in $Z_{p^{2} q r}$ tiles, where $p$ , $q$ and $r$ are primes. Combining this with a recent result of Malikiosis we obtain that Fuglede's conject…

math.CA2018★ 22 cited

On the discrete Fuglede and Pompeiu problems

Gergely Kiss, Romanos Diogenes Malikiosis, Gábor Somlai +1

We investigate the discrete Fuglede's conjecture and Pompeiu problem on finite abelian groups and develop a strong connection between the two problems. We give a geometric conditio…

Tilings of symmetric and alternating groups by conjugacy classes

Non-existence of sets with few special directions

Fuglede's Conjecture on Cyclic Groups of Square-Free Order: The Case of Rapidly Growing Prime Factors

Dihedral groups of square-free order are DCI-groups

Lower bounds for mask polynomials with many cyclotomic divisors

Functional tilings and the Coven-Meyerowitz tiling conditions

Spectrality of the product does not imply that the components are spectral

A lonely weak tile

On polynomials of small range sum

Spectral Tile Direction in the Group Zp2​×Zq2​×Zr​

Decompositions of the positive real numbers into disjoint sets closed under addition and multiplication

A new proof of Rédei's theorem on the number of directions

Tiling and weak tiling in (Zp​)d

Optimal embedded and enclosing isosceles triangles

On the diameter of Cayley graphs of classical groups with generating sets containing a transvection

CI-property of Cp2​×Cn​ and Cp2​×Cq2​ for digraphs

Special directions on the finite affine plane

Spectral sets and tiles in Zp2​×Zq2​

Fuglede's conjecture holds for cyclic groups of order pqrs

Minimum area isosceles containers

Fuglede's conjecture holds on Zp2​×Zq​

The Cayley isomorphism property for Zp3​×Zq​

Spectral sets in Zp2qr​ tile

On the discrete Fuglede and Pompeiu problems

Spectral Tile Direction in the Group $Z_{p^{2}} \times Z_{q^{2}} \times Z_{r}$

Tiling and weak tiling in $(Z_{p})^{d}$

CI-property of $C_{p}^{2} \times C_{n}$ and $C_{p}^{2} \times C_{q}^{2}$ for digraphs

Spectral sets and tiles in $Z_{p}^{2} \times Z_{q}^{2}$

Fuglede's conjecture holds for cyclic groups of order $pq r s$

Fuglede's conjecture holds on $Z_{p}^{2} \times Z_{q}$

The Cayley isomorphism property for $Z_{p}^{3} \times Z_{q}$

Spectral sets in $Z_{p^{2} q r}$ tile