Related papers: Shapiro's problem on polynomials with large partia…

For a fixed unit vector $a=(a_1,a_2,\ldots,a_n)\in S^{n-1}$, we consider the $2^n$ sign vectors $\varepsilon=(\varepsilon^1,\varepsilon^2,\ldots,\varepsilon^n)\in \{+1,-1\}^n$ and the corresponding scalar products $\varepsilon\cdot…

Schur's partition theorem states that the number of partitions of n into distinct parts congruent 1, 2 (mod 3) equals the number of partitions of n into parts which differ by >= 3, where the inequality is strict if a part is a multiple of…

We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemer\'{e}di theorem for distinct-degree polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of distinct degrees and…

Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review…

In this paper, we obtain a sharp upper bound for the sum of the first $k$-th eigenvalues for this Dirichlet problem of poly-Laplacian with any order, which is viewed as an extension of the result due to Cheng and Wei (Journal of…

By extending a construction due to Gross and McMullen [2], we show that for any odd integer n and for any even integer d>n+2 there are infinitely many Salem numbers $\alpha$ of degree d such that $\alpha^n-1$ is a unit. A similar result is…

Hardy showed that $\sum_{n \ioe x}\tau(n)-x(\log x +2\gamma -1)$ is not $o(x^{1/4})$. In this article, we prove that $\sum_{n \ioe x}\tau(n)(1-\frac{x}{n})-xP(\log x)=\frac{1}{4}+O \left( \frac{\log x}{x^{1/4}} \right)$, where $P$ is a…

We resolve two open problems on Kronecker coefficients $g(\lambda,\mu,\nu)$ of the symmetric group. First, we prove that for partitions $\lambda,\mu,\nu$ with fixed Durfee square size, the Kronecker coefficients grow at most polynomially.…

We extend the best known bound on the largest subset of {1,2,...,N} with no square differences to the largest possible class of quadratic polynomials.

Let $p(n)$ denote the number of partitions of a natural number $n$. As $ n \to \infty$, the $n$th root of $p(n)$ tends to $1$, which is related to the Cauchy--Hadamard test for power series. Andrews also discovered an elementary proof. Sun…

For any positive integers $n\ge d+1\ge 3$, what is the maximum size of a $(d+1)$-uniform set system in $[n]$ with VC-dimension at most $d$? In 1984, Frankl and Pach initiated the study of this fundamental problem and provided an upper bound…

Consider any Dirichlet series sum a_n/n^z with nonnegative coefficients a_n and finite sum function f(z)=f(x+iy) when x is greater than 1. Denoting the partial sum a_1+...+a_N by s_N, the paper gives the following necessary and sufficient…

Consider a quadratic polynomial $f\left(\xi_{1},\dots,\xi_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem,…

Let $c$ be a positive odd integer and $R$ a set of $n$ primes coprime with $c$. We consider equations $X + Y = c^z$ in three integer unknowns $X$, $Y$, $z$, where $z > 0$, $Y > X > 0$, and the primes dividing $XY$ are precisely those in…

Given a rational $a=p/q$ and $N$ nonnegative $d$-dimensional real vectors $u_1$, ..., $u_N$, we show that it is always possible to choose $(d-1)+\lceil (pN-d+1)/q\rceil$ of them such that their sum is (componentwise) at least…

We investigate a variant of Wirsing's problem on approximation to a real number by real algebraic numbers of degree exactly $n$. This has been studied by Bugeaud and Teulie. We improve their bounds for degrees up to $n=7$. Moreover, we…

A $d$-dimensional polycube is a facet-connected set of cells (cubes) on the $d$-dimensional cubical lattice $\mathbb{Z}^d$. Let $A_d(n)$ denote the number of $d$-dimensional polycubes (distinct up to translations) with $n$ cubes, and…

Given an odd integer polynomial f(x) of a degree k >=3, we construct a non-negative valued, normed trigonometric polynomial with the spectrum in the set of integer values of f(x) not greater than n, and a small free coefficient…

The paper studies the probability for a Galois group of a random polynomial to be $A_n$. We focus on the so-called large box model, where we choose the coefficients of the polynomial independently and uniformly from $\{-L,\ldots, L\}$. The…

A classical theorem of Nisan and Szegedy says that a boolean function with degree $d$ as a real polynomial depends on at most $d2^{d-1}$ of its variables. In recent work by Chiarelli, Hatami and Saks, this upper bound was improved to $C…