arXiv subjects
Publications and source records attributed to S. V. Konyagin.
For a fixed integer $n\ge 2,$ we consider the homogeneous polynomial $$ P(x_1, x_2, \ldots, x_{n+2})=\sum_{i=1}^{n} (x_2-x_1)^{i-1} x_1^{n-i} x_{i+2}. $$ We prove that, for any finite set $A$ of complex numbers, $$ \Bigl|\bigl\{P(x_1,x_2,\ldots,x_{n+2}): \, x_i\in A\bigr\}\Bigr|\gg |A|^{n}. $$ The implicit constant in $\gg$ may depend only on $n.$
Representative examples of our results are as follows. For any positive integer $N$ the equation $$ x^3+y^3=z^3+t^3, \quad x,y,z,t\in \mathbb{N}, \quad \{x,y\}\not=\{z,t\} $$ has no solutions satisfying $$ N\le x,y,z,t < N+\Bigl(\frac{38}{3}N+\frac{1297}{36}\Bigr)^{1/2}+\frac{19}{6}. $$ The strict inequality ``$<$" can not be substituted by ``$\le$", that is, there exist infinitely many positive integers $N$ such that the equation has a solution with $$ N\le x,y,z,t \le N+\Bigl(\frac{38}{3}N+\frac{1297}{36}\Bigr)^{1/2}+\frac{19}{6}. $$ There is an absolute constant $c>0$ such that for any positive integer $N$ the equation has a solution satisfying $$ N\le x,y,z,t \le N+cN^{2/3}. $$ For any $\varepsilon>0$ there exist infinitely many positive integers $N$ such that the equation has no solutions satisfying $$ N\le x,y,z,t \le N+N^{4/7-\varepsilon}. $$ There is an absolute constant $c>0$ such that for any positive integer $N$ the equation $$ x^4+y^4=z^4+t^4,\quad x,y,z,t\in\mathbb{N}, \quad \{x,y\}\not=\{z,t\}, $$ has no solutions satisfying $$ N\le x,y,z,t \le N+cN^{3/5}. $$ There is an absolute constant $c>0$ such that for any positive integer $N$ this equation has a solution satisfying $$ N\le x,y,z,t \le N+cN^{12/13}. $$
Let $\mathcal S=\{s_1<s_2<s_3<\ldots\}$ be the sequence of all natural numbers which can be represented as a sum of two squares of integers. For $X\ge2$ we denote by $g(X)$ the largest gap between consecutive elements of $\mathcal S$ that do not exceed $X$. We prove that for $X \to +\infty$ the lower bound $$g(X)\geq \left(\frac{390}{449}-o(1)\right)\ln X$$ holds. This estimate is twice the recent estimate by R. Dietmann and C. Elsholtz.
We sharpen the bounds of J. Bourgain, A. Gamburd and P. Sarnak (2016) on the possible number of nodes outside the "giant component" and on the size of individual connected components in the suitably defined functional graph of Markoff triples modulo $p$. This is a step towards the conjecture that there are no such nodes at all.
We prove that there exists an absolute constant $c>0$ such that if an arithmetic progression $\cP$ modulo a prime number $p$ does not contain zero and has the cardinality less than $cp$, then it can not be represented as a product of two subsets of cardinality greater than 1, unless $\cP=-\cP$ or $\cP=\{-2r,r,4r\}$ for some residue $r$ modulo $p$.
A continuous quadratic form ("quadratic form", in short) on a Banach space $X$ is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding symmetric linear operator $T\colon X\to X^*$ factors through a Hilbert space; (b) delta-convex (i.e., representable as a difference of two continuous convex functions) if and only if $T$ is a UMD-operator. It follows, for instance, that each quadratic form on an infinite-dimensional $L_p(μ)$ space ($1\le p \le\infty$) is: (a) delta-semidefinite iff $p \ge 2$; (b) delta-convex iff $p>1$. Some other related results concerning delta-convexity are proved and some open problems are stated.
Let $F_p$ be the field of a prime order $p$. Then for any positive integer $n>1$, for any $ε>0$, and for any subset $A$ of $F_p$, every element of $F_p$ can be represented as a sum of $N$ elements, each of them is a product of $n$ elements from $A$, where $N$ depends on $n$ and $\espilon$.
Let $τ(n)$ be the Ramanujan $τ$-function. We prove that for any integer $N$ the diophantine equation $$ \sum_{i=1}^{74000}τ(n_i)=N $$ has a solution in positive integers $n_1, n_2,..., n_{74000}$ satisfying the condition $$ \max_{1\le i\le 74000}n_i\ll |N|^{2/11}+1. $$ We also consider similar questions in the residue ring modulo a large prime $p.$
Let q be a prime, A be a subset of a finite field $F=\Bbb Z/q\Bbb Z$, $|A|<\sqrt{|F|}$. We prove the estimate $\max(|A+A|,|A\cdot A|)\ge c|A|^{1+ε}$ for some $ε>0$ and c>0. This extends the result of J. Bourgain, N. Katz, and T. Tao.
The paper is related to the following question of P.~L.~Ul'yanov: is it true that for any $2π$-periodic continuous function $f$ there is a uniformly convergent rearrangement of its trigonometric Fourier series? In particular, we give an affirmative answer if the absolute values of Fourier coefficients of $f$ decrease. Also, we study a problem how to choose $m$ terms of a trigonometric polynomial of degree $n$ to make the uniform norm of their sum as small as possible.