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Sergei V. Konyagin

Publications and source records attributed to Sergei V. Konyagin.

At least 19 recordsLinked to original sources
arXiv2025

Unique expansions in number systems via refinement equations

Using the subdivision schemes theory, we develop a criterion to check if any natural number has at most one representation in the $n$-ary number system with a set of non-negative integer digits $A=\{a_1, a_2,\ldots, a_n\}$ that contains zero. This uniqueness property is shown to be equivalent to a certain restriction on the roots of the trigonometric polynomial $\sum_{k=1}^n e^{-2πi a_k t}$. From this criterion, under a natural condition of irreducibility for $A$, we deduce that in case of prime $n$ the uniqueness holds if and only if the digits of $A$ are distinct modulo $n$, whereas for any composite $n$ we show that the latter condition is not necessary. We also establish the connection of this uniqueness to the semigroup freeness problem for affine integer functions of equal integer slope; this together with the two criteria allows to fill the gap in the work of D. Klarner on the question of P. Erdös about densities of affine integer orbits and establish a simple algorithm to check the freeness and the positivity of density when the slope is a prime number.

math.NT↗
arXiv2025

On Distinct Angles in the Plane

We prove that if $N$ points lie in convex position in the plane then they determine $Ω(N^{5/4})$ distinct angles, provided that the points do not lie on a common circle. This is derived from a more general claim that if $N$ points in the convex position in the real plane determine $KN$ distinct angles, then $K=Ω(N^{1/4})$ or $Ω(N/K)$ points are co-circular. The proof makes use of the implicit order one can give to points in convex position and relies on a slightly more general order assumption. The assumption enables one to reduce the issue to counting incidences between points and a multiset of cubic curves, with special attention being paid to the case when the curves are reducible.

math.CO↗
arXiv2024

Trigonometric polynomials with frequencies in the set of cubes

We prove that for any $ε>0$ and any trigonometric polynomial $f$ with frequencies in the set $\{n^3: N \leq n\leq N+N^{2/3-ε}\}$, one has $$ \|f\|_4 \ll ε^{-1/4}\|f\|_2 $$ with implied constant being absolute. We also show that the set $\{n^3: N\leq n\leq N+(0.5N)^{1/2}\}$ is a Sidon set.

math.CA↗
arXiv2023

Karatsuba's divisor problem and related questions

We prove that $$ \sum_{p \leq x} \frac{1}{τ(p-1)} \asymp \frac{x}{(\log x)^{3/2}}, \quad \quad \sum_{n \leq x} \frac{1}{τ(n^2+1)} \asymp \frac{x}{(\log x)^{1/2}}, $$ where $τ(n)=\sum_{d|n}1$ is the number of divisors of $n$, and the summation in the first sum is over primes.

math.NT↗
arXiv2023

On the distribution of multiplicatively dependent vectors

In this paper, we study the distribution of multiplicatively dependent vectors. For example, although they have zero Lebesgue measure, they are everywhere dense both in $\mathbb{R}^n$ and $\mathbb{C}^n$. We also study this property in a more detailed manner by considering the covering radius of such vectors.

math.NT↗
arXiv2022

Distinct Angles in General Position

The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is Erdős' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the plane. Recent work has introduced bounds on a wide array of variants of this problem, inspired by similar variants in the distance setting. In this short note, we improve the best known upper bound for the minimum number of distinct angles formed by $n$ points in general position from $O(n^{\log_2(7)})$ to $O(n^2)$. Before this work, similar bounds relied on projections onto a generic plane from higher dimensional space. In this paper, we employ the geometric properties of a logarithmic spiral, sidestepping the need for a projection. We also apply this configuration to reduce the upper bound on the largest integer such that any set of $n$ points in general position has a subset of that size with all distinct angles. This bound is decreased from $O(n^{\log_2(7)/3})$ to $O(n^{1/2})$.

cs.CG↗
arXiv2020

Polynomial Equations in Subgroups and Applications

We obtain a new bound for the number of solutions to polynomial equations in cosets of multiplicative subgroups in finite fields, which generalises previous results of P. Corvaja and U. Zannier (2013). We also obtain a conditional improvement of recent results of J. Bourgain, A. Gamburd and P. Sarnak (2016) and S. V. Konyagin, S. V. Makarychev, I. E. Shparlinski and I. V. Vyugin (2019) on the structure of solutions to the reduction of the Markoff equation $x^2 + y^2 + z^2 = 3 x yz$ modulo a prime $p$.

math.NT↗
arXiv2015

Functional Graphs of Polynomials over Finite Fields

Given a function $f$ in a finite field ${\mathbb F}_q$ of $q$ elements, we define the functional graph of $f$ as a directed graph on $q$ nodes labelled by the elements of ${\mathbb F}_q$ where there is an edge from $u$ to $v$ if and only if $f(u) = v$. We obtain some theoretic estimates on the number of non-isomorphic graphs generated by all polynomials of a given degree. We then develop a simple and practical algorithm to test the isomorphism of quadratic polynomials that has linear memory and time complexities. Furthermore, we extend this isomorphism testing algorithm to the general case of functional graphs, and prove that, while its time complexity increases only slightly, its memory complexity remains linear. We exploit this algorithm to provide an upper bound on the number of functional graphs corresponding to polynomials of degree $d$ over ${\mathbb F}_q$. Finally, we present some numerical results and compare function graphs of quadratic polynomials with those generated by random maps and pose interesting new problems.

math.NT↗
arXiv2014

Counting Additive Decompositions of Quadratic Residues in Finite Fields

We say that a set $S$ is additively decomposed into two sets $A$ and $B$ if $S = \{a+b : a\in A, \ b \in B\}$. A. Sárközy has recently conjectured that the set $Q$ of quadratic residues modulo a prime $p$ does not have nontrivial decompositions. Although various partial results towards this conjecture have been obtained, it is still open. Here we obtain a nontrivial upper bound on the number of such decompositions.

math.NT↗
arXiv2014

Quantitative version of Beurling-Helson theorem

It is proved that any continuous function f on the unit circle such that the sequence e^{in f}, n=1,2,... has small Wiener norm \| e^{in f} \|_A = o (\frac{\log^{1/22} |n|}{(\log \log |n|)^{3/11}}), is linear. Moreover, we get lower bounds for Wiener norm of characteristic functions of subsets from Z_p in the case of prime p.

math.CA↗
arXiv2013

Quadratic Non-residues in Short Intervals

We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes $p$ in a dyadic interval $[Q,2Q]$ for which a given interval $[u+1,u+ψ(Q)]$ does not contain a quadratic non-residue modulo $p$. The bound is nontrivial for any function $ψ(Q)\to\infty$ as $Q\to\infty$. This is an analogue of the well known estimates on the smallest quadratic non-residue modulo $p$ on average over primes $p$, which corresponds to the choice $u=0$.

math.NT↗
arXiv2013

New sum product type estimates

New lower bounds involving sum, difference, product, and ratio sets for a set $A\subset \C$ are given. The estimates involving the sum set match, up to constants, the one obtained by Solymosi for the reals and are obtained by generalising his approach to the complex plane. The bounds involving the difference set are slightly weaker. They improve on the best known ones, including the case $A\subset \R$, which also due to Solymosi, by means of combining the use of the Szemerédi-Trotter theorem with an arithmetic combinatorics technique.

math.CO↗
arXiv2012

Multiplicative Congruences with Variables from Short Intervals

Recently, several bounds have been obtained on the number of solutions to congruences of the type $$ (x_1+s)...(x_ν+s)\equiv (y_1+s)...(y_ν+s)\not\equiv0 \pmod p $$ modulo a prime $p$ with variables from some short intervals. Here, for almost all $p$ and all $s$ and also for a fixed $p$ and almost all $s$, we derive stronger bounds. We also use similar ideas to show that for almost all primes, one can always find an element of a large order in any rather short interval.

math.NT↗
arXiv2011

On the Hidden Shifted Power Problem

We consider the problem of recovering a hidden element $s$ of a finite field $\F_q$ of $q$ elements from queries to an oracle that for a given $x\in \F_q$ returns $(x+s)^e$ for a given divisor $e\mid q-1$. We use some techniques from additive combinatorics and analytic number theory that lead to more efficient algorithms than the naive interpolation algorithm, for example, they use substantially fewer queries to the oracle.

cs.CC↗
arXiv2011

Distribution of Elements of Cosets of Small Subgroups and Applications

We obtain a series of estimates on the number of small integers and small order Farey fractions which belong to a given coset of a subgroup of order $t$ of the group of units of the residue ring modulo a prime $p$, in the case when $t$ is small compared to $p$. We give two applications of these results: to the simultaneous distribution of two high degree monomials $x^{k_1}$ and $x^{k_2}$ modulo $p$ and to a question of J. Holden and P. Moree on fixed points of the discrete logarithm.

math.NT↗
arXiv2010

On the Convex Hull of the Points on Modular Hyperbolas

Given integers $a$ and $m\ge 2$, let $\Hm$ be the following set of integral points $$ \Hm= \{(x,y) \ : \ xy \equiv a \pmod m,\ 1\le x,y \le m-1\} $$ We improve several previously known upper bounds on $v_a(m)$, the number of vertices of the convex closure of $\Hm$, and show that uniformly over all $a$ with $\gcd(a,m)=1$ we have $v_a(m) \le m^{1/2 + o(1)}$ and furthermore, we have $v_a(m) \le m^{5/12 + o(1)}$ for $m$ which are almost squarefree.

math.NT↗
arXiv2010

Prime chains and Pratt trees

We study the distribution of prime chains, which are sequences p_1,...,p_k of primes for which p_{j+1}\equiv 1\pmod{p_j} for each j. We give estimates for the number of chains with p_k\le x (k variable), and the number of chains with p_1=p and p_k \le px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with p_k=p, which is also the height of the Pratt tree for p. We show H(p)\ge c\log\log p and H(p)\le (\log p)^{1-c'} for almost all p, with c,c' explicit positive constants. We can take, for any ε>0, c=e-εassuming the Elliott-Halberstam conjecture. A stochastic model of the Pratt tree, based on a branching random walk, is introduced and analyzed. The model suggests that for most p, H(p) stays very close to e \log\log p.

math.NT↗