Papers by Sergei Konyagin
Archiv Der Mathematik, 2007
We prove that for every integer N the Diophantine equation
Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a q... more Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.
Analytic Number Theory, 2015
For any positive integer k, we show that infinitely often, perfect k-th powers appear inside very... more For any positive integer k, we show that infinitely often, perfect k-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size
It is also shown that if $G$ is an abelian group of exponent 2, then there is a subset $S\subsete... more It is also shown that if $G$ is an abelian group of exponent 2, then there is a subset $S\subseteq G$ such that every element of $G$ has a representation as a sum of two elements of $S$, and the number of representations of non-zero elements is bounded by an absolute constant.
We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes... more 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) → ∞ as Q → ∞. 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.
Our first result is a 'sum-product' theorem for subsets A of the finite field Fp, p prime, provid... more Our first result is a 'sum-product' theorem for subsets A of the finite field Fp, p prime, providing a lower bound on max(|A + A|, |A · A|). The second and main result provides new bounds on exponential sums
Siam Journal on Computing, Dec 6, 2012
We consider the problem of recovering a hidden element s of a finite field F q of q elements from... more 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 ∈ F q returns (x + s) e for a given divisor e | 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.
Can J Math, 2009
Another application is to answer a question of Gowers. If B in Z/pZ is a set of size (p-1)/2 then... more Another application is to answer a question of Gowers. If B in Z/pZ is a set of size (p-1)/2 then there is some x in Z/pZ such that the intersection of B with B + x has cardinality within o(p) of p/4.
Eprint Arxiv Math 0605549, Apr 29, 2006
A continuous quadratic form ("quadratic form", in short) on a Banach space is called: (a) delta-s... more A continuous quadratic form ("quadratic form", in short) on a Banach space is called: (a) delta-semidefinite if it is the difference of two nonnegative quadratic forms; (b) delta-convex if it is the difference of two continuous convex functions. We characterize delta-semidefinite quadratic forms in terms of factorability of operators, and we construct a quadratic form on $\ell_1$ that is not delta-convex. It follows, for instance, that each quadratic form on an infinite-dimensional $L_p(\mu)$ space ($1\le p < \infty$) 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.
Real Analysis Exchange, 2002
ABSTRACT
Given a function $f$ in a finite field ${\mathbb F}_q$ of $q$ elements, we define the functional ... more 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.
Dedicated to Andrzej Schinzel on his sixtieth birthday
New lower bounds involving sum, difference, product, and ratio sets for a set $A\subset \C$ are g... more 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 Szemer\'edi-Trotter theorem with an arithmetic combinatorics technique.
We give a lower bound for Wiener norm of characteristic function of subsets
Ukrainian Mathematical Journal, 2010
In a linear space of dimension n over the field F 2 ; we construct a set A of given density such ... more In a linear space of dimension n over the field F 2 ; we construct a set A of given density such that the Fourier transform of A is large on a large set, and the intersection of A with any subspace of small dimension is small. The results obtained show, in a certain sense, the sharpness of one theorem of J. Bourgain.
Let τ (n) be the Ramanujan τ -function. We prove that for any integer N the diophantine equation ... more Let τ (n) be the Ramanujan τ -function. We prove that for any integer N the diophantine equation 74000 i=1
Given integers a and m ≥ 2, let H a (m) be the following set of integral points
We discuss three problems of the following kind: given a set A ⊆ F p of n := |A| residues modulo ... more We discuss three problems of the following kind: given a set A ⊆ F p of n := |A| residues modulo a prime p, how are the absolute values |S A (z)| of the corresponding exponential sums S A (z) := a∈A e 2πi az p ; z ∈ F p distributed in the interval [0, n]?
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Papers by Sergei Konyagin