arXiv subjects

Gergely Kiss

Publications and source records attributed to Gergely Kiss.

At least 19 recordsLinked to original sources

arXiv2026

N-ary quasi-arithmetic means and families without regularity

The classical theorems of Kolmogorov--Nagumo--de Finetti and of Aczel--Maksa characterize quasi-arithmetic means from two complementary directions: the former for compatible families of means satisfying the replacement axiom, and the latter for bisymmetric means of fixed arity. We refine both representation results by showing that the required continuity follows automatically. Our main result states that every reflexive, symmetric, bisymmetric and partially strictly increasing $n$-variable operation on a real interval is continuous and hence quasi-arithmetic. The proof is based on a recursive construction on $n$-adic rationals given by bisymmetry, and a dense-domain continuity argument. The same method also yields the regularity-free Kolmogorov--Nagumo--de Finetti theorem for compatible families of strictly increasing symmetric means.

math.GM↗

arXiv2026

On noncontinuous bisymmetric strictly monotone operations

We construct bisymmetric, strictly increasing binary operations on real intervals which are not continuous. This answers a natural question in the theory of bisymmetric and mean-type operations by showing that continuity may fail for non-reflexive operations of the form \[ F(x,y)=f^{-1}(\alpha f(x)+\beta f(y)), \] where $\alpha,\beta>0$ with $\alpha+\beta\neq1$. Our construction is based on a Cantor-type perfect set whose elements are linearly independent over a countable subfield of $\R$, which allows the generating function $f$ to map an interval bijectively onto a nowhere dense fractal-type set. As a consequence we obtain a noncontinuous associative and strictly increasing operation on an interval. We also extend the construction to the multivariate case. In the opposite direction we prove that if a symmetric bisymmetric strictly increasing operation is reflexive at two points of an interval, then it is automatically continuous on the segment between them and coincides there with a quasi-arithmetic mean.

math.GM↗

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

Signed null sequences and Hausdorff dimension

We investigate the convergence of signed null sequences of the form \[ \sum_{n=1}^\infty \varepsilon_n a_n, \quad \varepsilon_n \in \{-1,1\}, \] where $(a_n)$ tends to zero in $\mathbb{R}^d$. Our main result shows that for any such sequence, the set of sign sequences yielding convergence has full Hausdorff dimension in the natural ultrametric topology. This answers a question of Mattila in the one-dimensional case, for which we provide an elementary proof. Moreover, if $(a_n)\notin \ell^1$ in one dimension, then for every $L\in\mathbb{R}$ the set of sign sequences with sum $L$ also has Hausdorff dimension $1$. In higher dimensions the analogous statement does not hold in full generality, but it is guaranteed if the sequence has $d$ linearly independent L\'evy vectors.

math.CA↗

arXiv2025

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 $\Phi_{s_1},\dots,\Phi_{s_k}$ 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↗

arXiv2025

Invariants and equidecomposability in rings of polygons with sides of given directions

We investigate equidecomposability in the ring of polygons with sides restricted to given directions and using only translations. Extending classical results of Dehn and Hadwiger, we prove that equidecomposability in these rings is equivalent to the equality of some translation invariant simple valuations. We also consider the algebraic structure of direction sets. We show that under mild conditions, equidecomposability with respect to a set $S$ of slopes of the given directions is equivalent to equidecomposability with respect to the field generated by $S$. We also provide a complete description of all invariants of these polygon rings.

math.MG↗

arXiv2024

Isometric rigidity of the Wasserstein space over the plane with the maximum metric

We study $p$-Wasserstein spaces over the branching spaces $\mathbb{R}^2$ and $[-1,1]^2$ equipped with the maximum norm metric. We show that these spaces are isometrically rigid for all $p\geq1,$ meaning that all isometries of these spaces are induced by isometries of the underlying space via the push-forward operation. This is in contrast to the case of the Euclidean metric since with that distance the $2$-Wasserstein space over $\mathbb{R}^2$ is not rigid. Also, we highlight that the $1$-Wasserstein space is not rigid over the closed interval $[-1,1]$, while according to our result, its two-dimensional analog, the closed unit ball $[-1,1]^2$ with the more complicated geodesic structure is rigid.

math.MG↗

arXiv2024

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↗

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

Noise-Robust Keyword Spotting through Self-supervised Pretraining

Voice assistants are now widely available, and to activate them a keyword spotting (KWS) algorithm is used. Modern KWS systems are mainly trained using supervised learning methods and require a large amount of labelled data to achieve a good performance. Leveraging unlabelled data through self-supervised learning (SSL) has been shown to increase the accuracy in clean conditions. This paper explores how SSL pretraining such as Data2Vec can be used to enhance the robustness of KWS models in noisy conditions, which is under-explored. Models of three different sizes are pretrained using different pretraining approaches and then fine-tuned for KWS. These models are then tested and compared to models trained using two baseline supervised learning methods, one being standard training using clean data and the other one being multi-style training (MTR). The results show that pretraining and fine-tuning on clean data is superior to supervised learning on clean data across all testing conditions, and superior to supervised MTR for testing conditions of SNR above 5 dB. This indicates that pretraining alone can increase the model's robustness. Finally, it is found that using noisy data for pretraining models, especially with the Data2Vec-denoising approach, significantly enhances the robustness of KWS models in noisy conditions.

eess.AS↗

arXiv2024

Solutions to the discrete Pompeiu problem and to the finite Steinhaus tiling problem

Let $K$ be a nonempty finite subset of the Euclidean space $\mathbb{R}^k$ $(k\ge 2)$. We prove that if a function $f\colon \mathbb{R}^k\to \mathbb{C}$ is such that the sum of $f$ on every congruent copy of $K$ is zero, then $f$ vanishes everywhere. In fact, a stronger, weighted version is proved. As a corollary we find that every finite subset $K$ of $\mathbb{R}^k$ having at least two elements is a Jackson set; that is, no subset of $\mathbb{R}^k$ intersects every congruent copy of $K$ in exactly one point.

math.FA↗

arXiv2023

On polynomials of small range sum

In order to reprove an old result of R\'edei'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 Lov\'asz and Schrijver using discrete Fourier analysis.

math.NT↗

arXiv2023

Polynomial equations for additive functions I. The inner case

The aim of this sequence of work is to investigate polynomial equations satisfied by additive functions. As a result of this, new characterization theorems for homomorphisms and derivations can be given. More exactly, in this paper the following type of equation is considered $$\sum_{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x^{q_{i}})= 0 \qquad \left(x\in \mathbb{F}\right),$$ where $n$ is a positive integer, $\mathbb{F}\subset \mathbb{C}$ is a field, $f_{i}, g_{i}\colon \mathbb{F}\to \mathbb{C}$ are additive functions and $p_i, q_i$ are positive integers for all $i=1, \ldots, n$.

math.CA↗

arXiv2023

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↗

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

Polynomial equations for additive functions II

In this sequence of work we investigate polynomial equations of additive functions. We consider the solutions of equation \[ \sum_{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x)^{q_{i}}= 0 \qquad \left(x\in \mathbb{F}\right), \] where $n$ is a positive integer, $\mathbb{F}\subset \mathbb{C}$ is a field, $f_{i}, g_{i}\colon \mathbb{F}\to \mathbb{C}$ are additive functions and $p_i, q_i$ are positive integers for all $i=1, \ldots, n$. Using the theory of decomposable functions we describe the solutions as compositions of higher order derivations and field homomorphisms. In many cases we also give a tight upper bound for the order of the involved derivations. Moreover, we present the full description of the solutions in some important special cases, too.

math.CA↗

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↗

arXiv2023

Nonstandard $n$-distances based on certain geometric constructions

The concept of $n$-distance was recently introduced to generalize the classical definition of distance to functions of $n$ arguments. In this paper we investigate this concept through a number of examples based on certain geometrical constructions. In particular, our study shows to which extent the computation of the best constant associated with an $n$-distance may sometimes be difficult and tricky. It also reveals that two important graph theoretical concepts, namely the total length of the Euclidean Steiner tree and the total length of the minimal spanning tree constructed on $n$ points, are instances of $n$-distances.

math.MG↗