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Péter Maga

Publications and source records attributed to Péter Maga.

18 recordsLinked to original sources
arXiv2026

The non-spherical sup-norm problem for $\mathrm{GL}(n)$ and generalized spherical functions

We solve the sup-norm problem for minimal weight vectors in an arbitrary cuspidal representation $π$ of $\mathrm{GL}(n,\mathbb{Z})\backslash\mathrm{GL}(n,\mathbb{R})$ with a uniform power saving bound in terms of the archimedean data of $π$, including its spectral parameters and the dimension of its minimal $K$-type. As a key ingredient, we establish new uniform decay bounds for generalized spherical functions.

math.NT↗
arXiv2025

Extensions of integral orthoregular sets and icubes

In this paper, we study the question when a (rational or Gaussian) integral vector can be extended to an integral orthogonal basis consisting of vectors of equal length. We also study when a set of integral vectors has such an extension. Some necessary conditions are given which are proven to be sufficient in dimensions $3$ and $4$.

math.NT↗
arXiv2023

The sup-norm problem for automorphic cusp forms of $\mathrm{PGL}(n,\mathbb{Z}[i])$

Let $ϕ$ be an $L^2$-normalized Hecke--Maaß cusp form for $\mathrm{PGL}_n(\mathbb{Z}[i])$ on the locally symmetric space $X:=\mathrm{PGL}_n(\mathbb{Z}[i])\backslash \mathrm{PGL}_n(\mathbb{C}) / \mathrm{PU}_n$. If $Ω$ is a compact subset of $X$, then we prove the bound $\|ϕ|_Ω\|_{\infty}\ll_Ω λ_ϕ^{n(n-1)/4-δ}$ for some $δ>0$ depending only on $n$, where $λ_ϕ$ is the Laplace eigenvalue of $ϕ$.

math.NT↗
arXiv2023

The sequence of prime gaps is graphic

Let us call a simple graph on $n\geq 2$ vertices a prime gap graph if its vertex degrees are $1$ and the first $n-1$ prime gaps. We show that such a graph exists for every large $n$, and in fact for every $n\geq 2$ if we assume the Riemann hypothesis. Moreover, an infinite sequence of prime gap graphs can be generated by the so-called degree preserving growth process. This is the first time a naturally occurring infinite sequence of positive integers is identified as graphic. That is, we show the existence of an interesting, and so far unique, infinite combinatorial object.

math.CO↗
arXiv2022

Beyond the spherical sup-norm problem

We open a new perspective on the sup-norm problem and propose a version for non-spherical Maass forms when the maximal compact K is non-abelian and the dimension of the K-type gets large. We solve this problem for an arithmetic quotient of G=SL_2(C) with K=SU_2(C). Our results cover the case of vector-valued Maass forms as well as all the individual scalar-valued Maass forms of the Wigner basis, reaching sub-Weyl exponents in some cases. On the way, we develop analytic theory of independent interest, including uniform strong localization estimates for generalized spherical functions of high K-type and a Paley-Wiener theorem for the corresponding spherical transform acting on the space of rapidly decreasing functions. The new analytic properties of the generalized spherical functions lead to novel counting problems of matrices close to various manifolds that we solve optimally.

math.NT↗
arXiv2021

Generic power series on subsets of the unit disk

In this paper, we examine the boundary behaviour of the generic power series $f$ with coefficients chosen from a fixed bounded set $Λ$ in the sense of Baire category. Notably, we prove that for any open subset $U$ of the unit disk $D$ with a non-real boundary point on the unit circle, $f(U)$ is a dense set of $\CC$. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given.

math.CV↗
arXiv2020

The density hypothesis for horizontal families of lattices

We prove the density hypothesis for wide families of arithmetic orbifolds arising from all division quaternion algebras over all number fields of bounded degree. Our power-saving bounds on the multiplicities of non-tempered representations are uniform in the volume and spectral aspects.

math.NT↗
arXiv2020

Counting bounded elements of a number field

We estimate, in a number field, the number of elements and the maximal number of linearly independent elements, with prescribed bounds on their valuations. As a by-product, we obtain new bounds for the successive minima of ideal lattices. Our arguments combine group theory, ramification theory, and the geometry of numbers.

math.NT↗
arXiv2020

Analytic properties of spherical cusp forms on GL(n)

Let $ϕ$ be an $L^2$-normalized spherical vector in an everywhere unramified cuspidal automorphic representation of $\mathrm{PGL}_n$ over $\mathbb{Q}$ with Laplace eigenvalue $λ_ϕ$. We establish explicit estimates for various quantities related to $ϕ$ that are uniform in $λ_ϕ$. This includes uniforms bounds for spherical Whittaker functions on $\mathrm{GL}_n(\mathbb{R})$, uniform bounds for the global sup-norm of $ϕ$, and uniform bounds for the "essential support" of $ϕ$, i.e. the region outside which it decays exponentially. The proofs combine analytic and arithmetic tools.

math.NT↗
arXiv2018

The sup-norm problem for GL(2) over number fields

We solve the sup-norm problem for spherical Hecke-Maass newforms of square-free level for the group GL(2) over a number field, with a power saving over the local geometric bound simultaneously in the eigenvalue and the level aspect. Our bounds feature a Weyl-type exponent in the level aspect, they reproduce or improve upon all known special cases, and over totally real fields they are as strong as the best known hybrid result over the rationals.

math.NT↗
arXiv2018

The prime geodesic theorem in square mean

We strengthen the recent result of Cherubini and Guerreiro on the square mean of the error term in the prime geodesic theorem for $\mathrm{PSL}_2(\mathbb{Z})$. We also develop a short interval version of this result.

math.NT↗
arXiv2018

On the global sup-norm of GL(3) cusp forms

Let $ϕ$ be a spherical Hecke-Maass cusp form on the non-compact space $\mathrm{PGL}_3(\mathbb{Z})\backslash\mathrm{PGL}_3(\mathbb{R})$. We establish various pointwise upper bounds for $ϕ$ in terms of its Laplace eigenvalue $λ_ϕ$. These imply, for $ϕ$ arithmetically normalized and tempered at the archimedean place, the bound $\|ϕ\|_\infty\ll_ελ_ϕ^{39/40+ε}$ for the global sup-norm (without restriction to a compact subset). On the way, we derive a new uniform upper bound for the $\mathrm{GL}_3$ Jacquet-Whittaker function.

math.NT↗
arXiv2017

Random power series near the endpoint of the convergence interval

In this paper, we are going to consider power series $$ \sum_{n=1}^{\infty} a_nx^n, $$ where the coefficients $a_n$ are chosen independently at random from a finite set with uniform distribution. We prove that if the expected value of the coefficients is positive (resp. negative), then $$ \lim_{x\to 1-}\sum_{n=1}^{\infty} a_nx^n=\infty\qquad (\text{resp. }\lim_{x\to 1-}\sum_{n=1}^{\infty} a_nx^n=-\infty) $$ with probability $1$. Also, if the expected value of the coefficients is $0$, then $$ \limsup_{x\to 1-}\sum_{n=1}^{\infty} a_nx^n=\infty,\qquad \liminf_{x\to 1-}\sum_{n=1}^{\infty} a_nx^n=-\infty $$ with probability $1$. We investigate the analogous question in terms of Baire categories.

math.CA↗
arXiv2017

Applications of the Kuznetsov formula on GL(3): the level aspect

We develop an explicit Kuznetsov formula on GL(3) for congruence subgroups. Applications include a Lindelof on average type bound for the sixth moment of GL(3) L-functions in the level aspect, an automorphic large sieve inequality, density results for exceptional eigenvalues and density results for Maass forms violating the Ramanujan conjecture at finite places.

math.NT↗
arXiv2014

The sup-norm problem for PGL(4)

Let F be a Hecke-Maass cusp form for the group SL(4, Z) with Laplace eigenvalue lambda. Assume that F satisfies the Ramanujan conjecture at infinity (this is satisfied by almost all cusp forms). We show a power-saving sup-norm bound in terms of lambda (relative to the generic bound) for F restricted to a compact set.

math.NT↗
arXiv2012

A semi-adelic Kuznetsov formula over number fields

In these notes, we prove a semi-adelic version of the Kuznetsov formula over arbitrary number fields. The extent is the set of those automorphic vectors which are not necessarily spherical in the archimedean aspect and a class of weight functions which is important for applications.

math.NT↗
arXiv2012

How large dimension guarantees a given angle?

We study the following two problems: (1) Given $n\ge 2$ and $\al$, how large Hausdorff dimension can a compact set $A\su\Rn$ have if $A$ does not contain three points that form an angle $\al$? (2) Given $\al$ and $\de$, how large Hausdorff dimension can a %compact subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the $\de$-neighborhood of $\al$? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles 0 and $180^\circ$, the angles $60^\circ,90^\circ$ and $120^\circ$ also play special role in problem (2): the maximal dimension is smaller for these special angles than for the other angles. In problem (1) the angle $90^\circ$ seems to behave differently from other angles.

math.CA↗