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Fedor Petrov

Publications and source records attributed to Fedor Petrov.

35 records · Page 2Linked to original sources
arXiv2019

On small $n$-uniform hypergraphs with positive discrepancy

A two-coloring of the vertices $V$ of the hypergraph $H=(V, E)$ by red and blue has discrepancy $d$ if $d$ is the largest difference between the number of red and blue points in any edge. Let $f(n)$ be the fewest number of edges in an $n$-uniform hypergraph without a coloring with discrepancy $0$. Erdős and Sós asked: is $f(n)$ unbounded? N. Alon, D. J. Kleitman, C. Pomerance, M. Saks and P. Seymour proved upper and lower bounds in terms of the smallest non-divisor ($\mbox{snd}$) of $n$. We refine the upper bound as follows: $$f (n) \leq c \log \mbox{snd}\ {n}.$$

math.CO↗
arXiv2019

Hook inequalities

We give an elementary proof of the recent hook inequality given in [MPP3]: $\prod_{u\in λ} h(u) \, \le \, \prod_{u\in λ} h^\ast(u),$ where $h(u)$ is the usual hook in Young diagram $λ$, and $h^\ast(i,j)=i+j-1$. We then obtain a large variety of similar inequalities and their high-dimensional generalizations.

math.CO↗
arXiv2019

Congruences on sums of $q$-binomial coefficients

We establish a $q$-analogue of Sun--Zhao's congruence on harmonic sums. Based on this $q$-congruence and a $q$-series identity, we prove a congruence conjecture on sums of central $q$-binomial coefficients, which was recently proposed by Guo. We also deduce a $q$-analogue of a congruence due to Apagodu and Zeilberger from Guo's $q$-congruence.

math.NT↗
arXiv2018

Improved Bounds for Progression-Free Sets in $C_{8}^{n}$

Let $G$ be a finite group, and let $r_{3}(G)$ represent the size of the largest subset of $G$ without non-trivial three-term progressions. In a recent breakthrough, Croot, Lev and Pach proved that $r_{3}(C_{4}^{n}) \leqslant (3.61)^{n}$, where $C_{m}$ denotes the cyclic group of order $m$. For finite abelian groups $G \cong \prod_{i=1}^{n} C_{m_{i}}$, where $m_{1},\ldots,m_{n}$ denote positive integers such that $m_{1} | \ldots | m_{n}$, this also yields a bound of the form $r_{3}(G) \leqslant (0.903)^{\operatorname{rk}_{4}(G)} |G|$, with $\operatorname{rk}_{4}(G)$ representing the number of indices $i \in \left\{1,\ldots,n\right\}$ with $4\ |\ m_{i}$. In particular, $r_{3}(C_{8}^{n}) \leqslant (7.22)^{n}$. In this paper, we provide an exponential improvement for this bound, namely $r_{3}(C_{8}^{n}) \leq (7.09)^{n}$.

math.CO↗
arXiv2013

A new approach to constant term identities and Selberg-type integrals

Selberg-type integrals that can be turned into constant term identities for Laurent polynomials arise naturally in conjunction with random matrix models in statistical mechanics. Built on a recent idea of Karasev and Petrov we develop a general interpolation based method that is powerful enough to establish many such identities in a simple manner. The main consequence is the proof of a conjecture of Forrester related to the Calogero--Sutherland model. In fact we prove a more general theorem, which includes Aomoto's constant term identity at the same time. We also demonstrate the relevance of the method in additive combinatorics.

math.CO↗
arXiv2013

Combinatorial Nullstellensatz Approach to Polynomial Expanding

Applying techniques similar to Combinatorial Nullstellensatz we prove a lower estimate of $|f(A,B)|$ for finite subsets $A$, $B$ of a field, and polynomial $f(x,y)$ of the form $f(x,y)=g(x)+yh(x)$, where degree of $g$ is greater then degree of $h$.

math.CO↗
arXiv2013

General removal lemma

We formulate and prove a general result in spirit of hypergraph removal lemma for measurable functions of several variables.

math.CO↗
arXiv2013

Virtual continuity of the measurable functions of several variables, and Sobolev embedding theorems

Classical Luzin's theorem states that the measurable function of one variable is "almost" continuous. This is not so anymore for functions of several variables. The search of right analogue of the Luzin theorem leads to a notion of virtually continuous functions of several variables. This probably new notion appears implicitly in the statements like embeddings theorems and traces theorems for Sobolev spaces. In fact, it reveals their nature as theorems about virtual continuity. This notion is especially useful for the study and classification of measurable functions, aswell as in some questions on dynamical systems, polymorphisms and bistochastic measures. In this work we recall necessary definitions and properties of admissible metrics, define virtual continuity, describe some of applications. Detailed analysis is to be presented in another paper.

math.FA↗
arXiv2013

On Shirshov bases of graded algebras

We prove that if the neutral component in a finitely-generated associative algebra graded by a finite group has a Shirshov base, then so does the whole algebra.

math.RA↗
arXiv2012

Convex curves and a Poisson imitation of lattices

We solve a randomized version of the following open question: is there a strictly convex, bounded curve γin the plane such that the number of rational points on γ, with denominator $n$, approaches infinity with $n$? Although this natural problem appears to be out of reach using current methods, we consider a probabilistic analogue using a spatial Poisson-process that simulates the refined rational lattice $\frac{1}{d} Z^2$, which we call $M_d$, for each natural number $d$. The main result here is that with probability 1 there exists a strictly convex, bounded curve γsuch that the number of spatial Poisson points on γ, with intensity $d$, approaches infinity with $d$. The methods include the notion of a generalized affine length of a convex curve, defined by Petrov (2007).

math.MG↗
arXiv2011

On the interpolation of integer-valued polynomials

It is well known, that if polynomial with rational coefficients of degree $n$ takes integer values in points $0,1,...,n$ then it takes integer values in all integer points. Are there sets of $n+1$ points with the same property in other integral domains? We show that answer is negative for the ring of Gaussian integers $\mathbb{Z}[i]$ when $n$ is large enough. Also we discuss the question about minimal possible size of set, such that if polynomial takes integer values in all points of this set then it is integer-valued.

math.NT↗
arXiv2010

On possible growth of Toeplitz languages

We consider a new family of factorial languages whose subword complexity grows as $Θ(n^α)$, where $α$ is the root of some transcendent equation. Analytical methods and in particular, a corollary of the Wiener-Pitt theorem, are used to find the asymptotic growth of the complexity. Factorial languages considered are languages of arithmetical factors of some Toeplitz words. So, we describe a new family of words with an unusual growth of arithmetical complexity.

math.CO↗
arXiv2010

Frugal Mechanism Design via Spectral Techniques

We study the design of truthful mechanisms for set systems, i.e., scenarios where a customer needs to hire a team of agents to perform a complex task. In this setting, frugality [Archer&Tardos'02] provides a measure to evaluate the "cost of truthfulness", that is, the overpayment of a truthful mechanism relative to the "fair" payment. We propose a uniform scheme for designing frugal truthful mechanisms for general set systems. Our scheme is based on scaling the agents' bids using the eigenvector of a matrix that encodes the interdependencies between the agents. We demonstrate that the r-out-of-k-system mechanism and the \sqrt-mechanism for buying a path in a graph [Karlin et. al'05] can be viewed as instantiations of our scheme. We then apply our scheme to two other classes of set systems, namely, vertex cover systems and k-path systems, in which a customer needs to purchase k edge-disjoint source-sink paths. For both settings, we bound the frugality of our mechanism in terms of the largest eigenvalue of the respective interdependency matrix. We show that our mechanism is optimal for a large subclass of vertex cover systems satisfying a simple local sparsity condition. For k-path systems, while our mechanism is within a factor of k + 1 from optimal, we show that it is, in fact, optimal, when one uses a modified definition of frugality proposed in [Elkind et al.'07]. Our lower bound argument combines spectral techniques and Young's inequality, and is applicable to all set systems. As both r-out-of-k systems and single path systems can be viewed as special cases of k-path systems, our result improves the lower bounds of [Karlin et al.'05] and answers several open questions proposed in that paper.

cs.GT↗