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Gábor Somlai

Publications and source records attributed to Gábor Somlai.

At least 19 recordsLinked to original sources
arXiv2026

Tilings of symmetric and alternating groups by conjugacy classes

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 Rothaus and Thompson in 1966 and remains open. We prove that this long-standing case is the only unresolved case for symmetric groups: every other nonidentity conjugacy class of $S_n$ fails to tile $S_n$. Our proofs combine representation-theoretic methods with combinatorial arguments concerning products of permutations. We also completely resolve the corresponding problem for alternating groups in a stronger form: for $n\geq5$, no normal subset of $A_n$ that avoids the identity can tile $A_n$. \iffalse We study the problem of whether a conjugacy class tiles the symmetric group or alternating group. For $S_n$, the case where the conjugacy class consists of transpositions was first investigated by Rothaus and Thompson in 1966 and remains open. We prove that this long-standing case is the only unresolved case for symmetric groups: every other nonidentity conjugacy class of $S_n$ fails to tile $S_n$. The proof combines representation-theoretic methods with combinatorial arguments concerning products of permutations. We also completely resolve the corresponding problem for alternating groups in a stronger form: for $n\geq5$, no normal subset of $A_n$ that avoids the identity can tile $A_n$. \fi \iffalse In this paper, we study when a conjugacy class or a normal subset tiles the symmetric or alternating group. The case of transpositions was first investigated by Rothaus and Thompson, and in general, it remains open. We show that every other conjugacy class of a symmetric group does not tile. We also prove that normal subsets in $A_n$ ($n\ge 5)$ not containing the identity element cannot tile.

math.GR↗
arXiv2026

Non-existence of sets with few special directions

Let $p$ be an odd prime and let $S\subseteq\mathbb{F}_p^2$ have cardinality divisible by $p$. We prove that, for all sufficiently large primes $p$, no such set has exactly four special directions, and obtain a conditional extension to larger numbers of special directions under an affine-independence assumption on the corresponding projection functions. A separate second-moment argument shows more generally that, for every fixed $k\ge4$, no subset of $\mathbb{F}_p^2$ has exactly $k$ special directions once $p$ is sufficiently large. In fact, the result holds uniformly for $k$ up to a positive constant times $\sqrt p/\log p$. In contrast, for multisets every prescribed collection of at most $p$ directions can occur as the set of special directions of a $\{0,1,2\}$-valued multiset on $\mathbb{F}_p^2$.

math.NT↗
arXiv2026

Spectral Tile Direction in the Group $\mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2} \times \mathbb{Z}_r$

Let $p$, $q$, and $r$ be distinct primes such that $p^2q^2<r$. We prove that every spectral set in the cyclic group $\mathbb{Z}_{p^2q^2r}$ is a tile. Since the reverse direction is already known, this shows that $\mathbb{Z}_{p^2q^2r}$ is a Fuglede group under this condition. The proof is based on divisibility properties of mask polynomials and on the structure of spectral sets in finite cyclic groups.

math.CA↗
arXiv2026

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

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-spectral direction of the conjecture holds for all square-free cyclic groups, whereas we prove the spectral-tiling direction only for those groups among them whose prime factors grow rapidly. To achieve this, we develop an inductive technique with roots in a previous paper by the author co-authored with Fallon, Kiss, and Mayeli. Following recent preprints, Fuglede's conjecture remains open only for finite cyclic groups, and until now, there was no cyclic group for which the conjecture was known that possessed an arbitrary number of distinct divisors.

math.CO↗
arXiv2026

On polynomials of small range sum

In order to reprove an old result of Rédei's on the number of directions determined by a set of cardinality $p$ in $\mathbb{F}_p^2$, Somlai proved that the non-constant polynomials over the field $\mathbb{F}_p$ whose range sums are equal to $p$ are of degree at least $\frac{p-1}{2}$. Here the summand in the range sum are considered as integers from the interval $[0,p-1]$. In this paper we characterise all of these polynomials having degree exactly $\frac{p-1}{2}$, if $p$ is large enough. As a consequence, for the same set of primes we re-establish the characterisation of sets with few determined directions due to Lovász and Schrijver using discrete Fourier analysis.

math.NT↗
arXiv2026

Lower bounds for mask polynomials with many cyclotomic divisors

Given a nonempty set $A \subset \mathbb{N}\cup\{0\}$, 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 cyclotomic polynomials $Φ_{s_1},\dots,Φ_{s_k}$ divide $A(X)$. What is the smallest possible size of $A$? For $k=1$, this was answered by Lam and Leung in 2000. Less is known about the case when $k\geq 2$; in particular, one may ask whether (similarly to the $k=1$ case) the optimal configurations have a simple ``fibered" structure on each scale involved. We prove that this is true in a number of special cases, but false in general, even if further strong structural assumptions are added. Results of this type are expected to have a broad range of applications, including Favard length of product Cantor sets, Fuglede's spectral set conjecture, and the Coven-Meyerowitz conjecture on integer tilings.

math.NT↗
arXiv2026

Cylinder type and $p$-divisible sets in $\mathbb{F}_p^3$

A set of points $S \subseteq \mathbb{F}_p^n$ is called \emph{$p$-divisible} if every affine hyperplane in $\mathbb{F}_p^n$ intersects $S$ in $0 \pmod p$ points. The Strong Cylinder Conjecture of Ball asserts that if $S$ is a $p$-divisible set of $p^2$ points in $\mathbb{F}_p^3$, then $S$ is a cylinder. In this paper, we show that every $p$-divisible multiset $S$ is both a $\mathbb{F}_p$-linear and $\mathbb{Z}$-linear combination of characteristic functions of cylinders. In addition, the multisets of size $p^2$ are $\Z$-linear combinations of a plane and weighted differences of parallel lines.

math.CO↗
arXiv2025

Functional tilings and the Coven-Meyerowitz tiling conditions

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 conjecture is reduced to sets which tile a finite cyclic group $\mathbb{Z}_M$. In this paper we consider a natural relaxation of this problem, where we replace sets with nonnegative functions $f,g$, such that $f(0)=g(0)=1$, $f\ast g=\mathbf{1}_{\mathbb{Z}_M}$ is a functional tiling, and $f, g$ satisfy certain further natural properties associated with tilings. We show that the Coven-Meyerowitz tiling conditions do not necessarily hold in such generality. Such examples of functional tilings carry the potential to lead to proper tiling counterexamples to the Coven-Meyerowitz conjecture in the future.

math.CO↗
arXiv2025

Dihedral groups of square-free order are DCI-groups

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 DCI-group, then Dobson, Muzychuk, and Spiga proved that $G$ must be a dihedral group of square-free order (Ars Math. Contemp., 2022). In this paper, we prove that the converse statement also holds, i.\,e., all dihedral groups of square-free order are DCI-groups.

math.GR↗
arXiv2025

The generalized trifference problem

We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. For $m = 1$, this is the classical trifference problem which is wide open. We prove upper and lower bounds on $T(n, m)$ for various ranges of the parameter $m$ and determine the phase transition threshold on $m=m(n)$ where $T(n, m)$ jumps from constant to exponential in $n$. By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters.

math.CO↗
arXiv2024

A lonely weak tile

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 complement weakly with a suitable Borel measure. In this paper we review the concept of weak tiling, and answer a question raised in \cite{weak} by giving an example of a set $T$ which tiles its complement weakly, but $T$ is neither spectral, nor a proper tile.

math.CO↗
arXiv2024

Degree gap of polynomials of small range sum

Polynomials of degree $\frac{p-1}{2}$ of range sum $p$ were determined by the second authors relying on a joint work of the authors by Kiss and Nagy. We prove that for large enough primes there is no polynomial of degree $\frac{p+1}{2}$ of range sum $p$.

math.NT↗
arXiv2023

Spectral sets in $\mathbb{Z}_{p^2qr}$ tile

We prove the every spectral set in $\mathbb{Z}_{p^2qr}$ tiles, where $p$, $q$ and $r$ are primes. Combining this with a recent result of Malikiosis we obtain that Fuglede's conjecture holds for $\mathbb{Z}_{p^2qr}$.

math.CA↗
arXiv2023

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

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 multiplication. As a byproduct of the argument we determine all the possible decompositions of the transcendental extension of the rational field of rank one into two pieces. Further, we prove that the positive elements of a real algebraic extensions of the rational numbers are indecomposable into two pieces.

math.NT↗
arXiv2023

Special directions on the finite affine plane

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 no such a set with exactly two special directions. We characterise sets with exactly 3 special directions which answers a question of Ghidelli in negative. Further we introduce methods to construct sets of minimal cardinality that has exactly 4 special directions for small values of $p$.

math.CO↗
arXiv2022

Tiling and weak tiling in $(\mathbb{Z}_p)^d$

We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary $p$-groups $(\mathbb{Z}_p)^d$, we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for $d=1, 2$, and prove some partial results for $d=3$.

math.CO↗
arXiv2022

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

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 $Cay(G,X)$ is bounded by $\log|G|^c$ for some universal constant $c$. In this paper, we prove such a bound for $Cay(G,X)$ for $G=PSL(n,q),PSp(n,q)$ or $PSU(n,q)$ where $q$ is odd, under the assumptions that $X$ contains a transvection and $q\neq 9$ or $81$.

math.GR↗