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Akos Magyar

Publications and source records attributed to Akos Magyar.

At least 19 recordsLinked to original sources
arXiv2026

Simplex--center configurations in dense subsets of Euclidean spaces and the integer lattice

We obtain density Ramsey theorems for configurations consisting of the vertices of a simplex $\Delta_o$ together with their barycenter. We prove that any subset $A\subseteq\mathbb{R}^n$ of positive upper density contains an isometric copy of all sufficiently large dilates of $\Delta_o$ together with its barycenter. As this configuration is non-spherical such results are not possible with respect to the quadratic Euclidean metric, we consider general metrics $\rho$ defined by a positive-definite, homogeneous forms of even degree at least four. We prove the analogous result in the discrete setting, for subsets $A$ of the integer lattice $\mathbb{Z}^n$, under some natural and necessary congruence restrictions on the scales $\lambda$ at which the set $A$ can contain an isometric copy of the simplex.

math.MG↗
arXiv2026

On almost primes solutions to forms of odd degrees in many variables

Let $\mathcal{F}=\{f_1,\ldots,f_R\}$ be a family of forms of odd degrees at most $d$ in $s$ variables. We study the solutions to the system $f_1(\mathbf{x})=\ldots=f_R(\mathbf{x})=0$ of the form $x_i=y_ip_i$ with $|y_i|\leq Y_\mathcal{F}$ and $p_i$ being a prime for all $i\in [s]$ inside the box $[-N,N]^s$, for large $N$. We show that if the number of variables $s$ is sufficiently large with respect to the parameters $R$ and $d$, then there are at least $C_\mathcal{F} N^{s-D}/(\log\,N)^s$ such solutions for some constants $C_\mathcal{F}>0$ and $D\in\mathbb{N}$, with $D$ depending only on the initial parameters $R$ and $d$.

math.NT↗
arXiv2025

The VC-dimension and point configurations in $\mathbb{R}^d$

Given a set $X$ and a collection ${\mathcal H}$ of functions from $X$ to $\{0,1\}$, the VC-dimension measures the complexity of the hypothesis class $\mathcal{H}$ in the context of PAC learning. In recent years, this has been connected to geometric configuration problems in vector spaces over finite fields. In particular, it is easy to show that the VC-dimension of the set of spheres of a given radius in $\mathbb{F}_q^d$ is equal to $d+1$, since this is how many points generically determine a sphere. It is known that for $E\subseteq \mathbb{F}_q^d$, $|E|\geq q^{d-\frac{1}{d-1}}$, the set of spheres centered at points in $E$, and intersected with the set $E$, has VC-dimension either $d$ or $d+1$. In this paper, we study a similar question over Euclidean space. We find an explicit dimensional threshold $s_d<d$ so that whenever $E\subseteq \mathbb{R}^d$, $d\geq 3$, and the Hausdorff dimension of $E$ is at least $s_d$, it follows that there exists an interval $I$ such that for any $t\in I$, the VC-dimension of the set of spheres of radius $t$ centered at points in $E$, and intersected with $E$, is at least $3$. In the process of proving this theorem, we also provide the first explicit dimensional threshold for a set $E\subseteq \mathbb{R}^3$ to contain a $4$-cycle, i.e. $x_1,x_2,x_3,x_4\in E$ satisfying $$ |x_1-x_2|=|x_2-x_3|=|x_3-x_4|=|x_4-x_1| $$

math.CA↗
arXiv2025

On the frequency of small gaps between the primes

In a recent work Friedlander studied the problem of how large consecutive prime gaps should be in order that the sum of the reciprocals should be divergent. Supposing a very deep Hypothesis, a generalization of the Hardy--Littlewood prime $k$-tuple conjecture, he gave an almost precise answer for it. In the present work we give an unconditional answer for a much weaker form of the same problem.

math.NT↗
arXiv2023

Spherical Configurations over Finite Fields

We establish that if $d \geq 2k + 6$ and $q$ is odd and sufficiently large with respect to $α\in (0,1)$, then every set $A\subseteq \mathbf{F}_q^d$ of size $|A| \geq αq^d$ will contain an isometric copy of every spherical $(k+2)$-point configuration that spans $k$ dimensions.

math.CO↗
arXiv2023

Weak hypergraph regularity and applications to geometric Ramsey theory

Let $Δ=Δ_1\times\ldots\times Δ_d\subseteq\mathbb{R}^n$, where $\mathbb{R}^n=\mathbb{R}^{n_1}\times\cdots\times\mathbb{R}^{n_d}$ with each $Δ_i\subseteq\mathbb{R}^{n_i}$ a non-degenerate simplex of $n_i$ points. We prove that any set $S\subseteq \mathbb{R}^n$, with $n=n_1+\cdots +n_d$ of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of the configuration $Δ$. In particular any such set $S\subseteq \mathbb{R}^{2d}$ contains a $d$-dimensional cube of side length $λ$, for all $λ\geq λ_0(S)$. We also prove analogous results with the underlying space being the integer lattice. The proof is based on a weak hypergraph regularity lemma and an associated counting lemma developed in the context of Euclidean spaces and the integer lattice.

math.CO↗
arXiv2022

Polynomial sequences in discrete nilpotent groups of step 2

We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise convergence theorem for ergodic averages along polynomial sequences, and a nilpotent Waring theorem. Our proofs are based on analytical tools, such as a nilpotent Weyl inequality, and on complex almost-orthogonality arguments that are designed to replace Fourier transform tools, which are not available in the non-commutative nilpotent setting. In particular, we present what we call a "nilpotent circle method" that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.

math.CA↗
arXiv2020

Simplices in thin subsets of Euclidean spaces

Let $\De$ be a non-degenerate simplex on $k$ vertices. We prove that there exists a threshold $s_k<k$ such that any set $A\subs \R^k$ of Hausdorff dimension $dim\,A\geq s_k$ necessarily contains a similar copy of the simplex $\De$.

math.CA↗
arXiv2020

Multilinear maximal operators associated to simplices

We establish $L^{p_1}\times\cdots\times L^{p_k}\to L^r$ and $\ell^{p_1}\times\cdots\times \ell^{p_k}\to \ell^r$ type bounds for multilinear maximal operators associated to averages over isometric copies of a given non-degenerate $k$-simplex in both the continuous and discrete settings. These provide natural extensions of $L^p\to L^p$ and $\ell^p\to \ell^p$ bounds for Stein's spherical maximal operator and the discrete spherical maximal operator, with each of these results serving as a key ingredient of the respective proofs.

math.CA↗
arXiv2019

Distances and Trees in Dense Subsets of $\mathbb{Z}^d$

In \cite{FKW} Katznelson and Weiss establish that all sufficiently large distances can always be attained between pairs of points from any given measurable subset of $\mathbb{R}^2$ of positive upper (Banach) density. A second proof of this result, as well as a stronger "pinned variant", was given by Bourgain in \cite{B} using Fourier analytic methods. In \cite{M1} the second author adapted Bourgain's Fourier analytic approach to established a result analogous to that of Katznelson and Weiss for subsets $\mathbb{Z}^d$ provided $d\geq 5$. We present a new direct proof of this discrete distance set result and generalize this to arbitrary trees. Using appropriate discrete spherical maximal function theorems we ultimately establish the natural "pinned variants" of these results.

math.NT↗
arXiv2017

Product of simplices and sets of positive upper density in $\mathbb{R}^d$

We establish that any subset of $\mathbb{R}^d$ of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided $d\geq4$. We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if $Δ_{k_1}$ and $Δ_{k_2}$ are two fixed non-degenerate simplices of $k_1+1$ and $k_2+1$ points respectively, then any subset of $\mathbb{R}^d$ of positive upper Banach density with $d\geq k_1+k_2+6$ will necessarily contain an isometric copy of all sufficiently large dilates of $Δ_{k_1}\timesΔ_{k_2}$. A new direct proof of the fact that any subset of $\mathbb{R}^d$ of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of $k+1$ points provided $d\geq k+1$, a result originally due to Bourgain, is also presented.

math.CA↗
arXiv2017

Simplices and sets of positive upper density in $\mathbb{R}^d$

We prove an extension of Bourgain's theorem on pinned distances in measurable subset of $\mathbb{R}^2$ of positive upper density, namely Theorem $1^\prime$ in [Bourgain, 1986], to pinned non-degenerate $k$-dimensional simplices in measurable subset of $\mathbb{R}^{d}$ of positive upper density whenever $d\geq k+2$ and $k$ is any positive integer.

math.CA↗
arXiv2016

Almost prime solutions to diophantine systems of high rank

Let $\F$ be a family of $r$ integral forms of degree $k\geq 2$ and $\LL=(l_1,\ldots,l_m)$ be a family of pairwise linearly independent linear forms in $n$ variables $\x=(x_1,...,x_n)$. We study the number of solutions $\x\in[1,N]^n$ to the diophantine system $\F(\x)=\vv$ under the restriction that $l_i(\x)$ has a bounded number of prime factors for each $1\leq i\leq m$. We show that the system $\F$ have the expected number of such "almost prime" solutions under similar conditions as was established for existence of integer solutions by Birch.

math.NT↗
arXiv2010

On restricted arithmetic progressions over finite fields

Let A be a subset of $\F_p^n$, the $n$-dimensional linear space over the prime field $\F_p$ of size at least $\de N$ $(N=p^n)$, and let $S_v=P^{-1}(v)$ be the level set of a homogeneous polynomial map $P:\F_p^n\to\F_p^R$ of degree $d$, and $v\in\F_p^R$. We show, that under appropriate conditions, the set $A$ contains at least $c\, N|S|$ arithmetic progressions of length $l\leq d$ with common difference in $S_v$, where c is a positive constant depending on $\de$, $l$ and $P$. We also show that the conditions are generic for a class of sparse algebraic sets of density $\approx N^{-\eps}$.

math.NT↗