arXiv subjects

Angel Kumchev

Publications and source records attributed to Angel Kumchev.

10 recordsLinked to original sources

arXiv2023

Short Interval Results For Powerfree Polynomials Over Finite Fields

Let $k \geq 2$ be an integer and $\mathbb F_q$ be a finite field with $q$ elements. We prove several results on the distribution in short intervals of polynomials in $\mathbb F_q[x]$ that are not divisible by the $k$th power of any non-constant polynomial. Our main result generalizes a recent theorem by Carmon and Entin on the distribution of squarefree polynomials to all $k \ge 2$. We also develop polynomial versions of the classical techniques used to study gaps between $k$-free integers in $\mathbb Z$. We apply these techniques to obtain analogues in $\mathbb F_q[x]$ of some classical theorems on the distribution of $k$-free integers. The latter results complement the main theorem in the case when the degrees of the polynomials are of moderate size.

math.NT↗

arXiv2022

Explicit bounds for large gaps between squarefree integers

We obtain explicit forms of the current best known asymptotic upper bounds for gaps between squarefree integers. In particular we show, for any $x \ge 2$, that every interval of the form $(x, x + 11x^{1/5}\log x]$ contains a squarefree integer. The constant 11 can be improved further, if $x$ is assumed to be larger than a (very) large constant.

math.NT↗

arXiv2019

On the ergodic Waring--Goldbach problem

We prove an asymptotic formula for the Fourier transform of the arithmetic surface measure associated to the Waring--Goldbach problem and provide several applications, including bounds for discrete spherical maximal functions along the primes and distribution results such as ergodic theorems.

math.CA↗

arXiv2018

Improved $\ell^p$-Boundedness for Integral $k$-Spherical Maximal Functions

We improve the range of $\ell^p(\mathbb Z^d)$-boundedness of the integral $k$-spherical maximal functions introduced by Magyar. The previously best known bounds for the full $k$-spherical maximal function require the dimension $d$ to grow at least cubicly with the degree $k$. Combining ideas from our prior work with recent advances in the theory of Weyl sums by Bourgain, Demeter, and Guth and by Wooley, we reduce this cubic bound to a quadratic one. As an application, we deduce improved bounds in the ergodic Waring--Goldbach problem.

math.CA↗

arXiv2018

A hybrid of two theorems of Piatetski-Shapiro

Let $c > 1$ and $0 < γ< 1$ be real, with $c \notin \mathbb N$. We study the solubility of the Diophantine inequality \[ \left| p_1^c + p_2^c + \dots + p_s^c - N \right| < \varepsilon \] in Piatetski-Shapiro primes $p_1, p_2, \dots, p_s$ of index $γ$---that is, primes of the form $p = \lfloor m^{1/γ} \rfloor$ for some $m \in \mathbb N$.

math.NT↗

arXiv2016

On sums of powers of almost equal primes

Let $k \ge 2$ and $s$ be positive integers, and let $n$ be a large positive integer subject to certain local conditions. We prove that if $s \ge k^2+k+1$ and $θ> 31/40$, then $n$ can be expressed as a sum $p_1^k + \dots + p_s^k$, where $p_1, \dots, p_s$ are primes with $|p_j - (n/s)^{1/k}| \le n^{θ/k}$. This improves on earlier work by Wei and Wooley and by Huang who proved similar theorems when $θ> 19/24$.

math.NT↗

arXiv2011

Sums of almost equal squares of primes

We study the representations of large integers $n$ as sums $p_1^2 + ... + p_s^2$, where $p_1,..., p_s$ are primes with $| p_i - (n/s)^{1/2} | \le n^{θ/2}$, for some fixed $θ< 1$. When $s = 5$ we use a sieve method to show that all sufficiently large integers $n \equiv 5 \pmod {24}$ can be represented in the above form for $θ> 8/9$. This improves on earlier work by Liu, Lü and Zhan, who established a similar result for $θ> 9/10$. We also obtain estimates for the number of integers $n$ satisfying the necessary local conditions but lacking representations of the above form with $s = 3, 4$. When $s = 4$ our estimates improve and generalize recent results by Lü and Zhai, and when $s = 3$ they appear to be first of their kind.

math.NT↗

arXiv2009

Additive bases arising from functions in a Hardy field

A classical additive basis question is Waring's problem. It has been extended to integer polynomial and non-integer power sequences. In this paper, we will consider a wider class of functions, namely functions from a Hardy field, and show that they are asymptotic bases.

math.NT↗

arXiv2009

On sums of squares of primes II

In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious oversight in our first paper, but make further progress on the exponential sums estimates needed, together with an embellishment of the previous sieve technique employed. This leads to an improvement in our bounds for the maximal size of the exceptional sets.

math.NT↗

arXiv2004

On the irreducibility of a truncated binomial expansion

Let P_nk(x) denote the sum of the lowest k+1 terms in the expansion of (1+x)^n. We investigate the irreducibility of P_nk(x) and more general univariate polynomials related to it. Polynomials P_nk(x) naturally arise in Schubert calculus, see ArXiv e-print math.RT/0409329 by I.Scherbak.

math.NT↗