Related papers: On transcendental entire functions with infinitely…

We show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ holds almost everywhere for all $f \in H^s (\mathbb{R}^n)$ provided that $s>\frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to the endpoint, this result is…

Evaluating or finding the roots of a polynomial $f(z) = f_0 + \cdots + f_d z^d$ with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of $f$ obtained with a careful use of the Newton polygon of…

Motivated by an influential result of Bourgain and Tzafriri, we consider continuous matrix functions $A:\mathbb{R}\to M_{n\times n}$ and lower $\ell_2$-norm bounds associated with their restriction to certain subspaces. We prove that for…

Let $R$ be a commutative Noetherian ring, $I$ an ideal, $M$ and $N$ finitely generated $R$-modules. Assume $V(I)\cap Supp(M)\cap Supp(N)$ consists of finitely many maximal ideals and let ${\l}(\e^i(N/I^nN,M))$ denote the length of…

Let $\{a_n\}_{n\in\mathbb{N}}$, $\{b_n\}_{n\in \mathbb{N}}$ be two infinite subsets of positive integers and $\psi:\mathbb{N}\to \mathbb{R}_{>0}$ be a positive function. We completely determine the Hausdorff dimensions of the set of all…

Consider the problem of minimizing a lower semi-continuous semi-algebraic function $f \colon \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ on an unbounded closed semi-algebraic set $S \subset \mathbb{R}^n.$ Employing adequate tools of…

For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the large-$q$ limit, we…

We determine a lower gap property for the growth of an unbounded \(\mathbb{Z}\)-valued \(k\)-regular sequence. In particular, if \(f:\mathbb{N}\to\mathbb{Z}\) is an unbounded \(k\)-regular sequence, we show that there is a constant \(c>0\)…

In this note, the main focus is on a question about transcendental entire functions mapping $\mathbb{Q}$ into $\mathbb{Q}$ (which is related to a Mahler's problem). In particular, we prove that, for any $t>0$, there is no a transcendental…

Let $X_N$ be an $N$-dimensional subspace of $L_2$ functions on a probability space $(\Omega, \mu)$ spanned by a uniformly bounded Riesz basis $\Phi_N$. Given an integer $1\leq v\leq N$ and an exponent $1\leq q\leq 2$, we obtain universal…

On the one hand, it is well known that the only subquadratic Dehn function of finitely presented groups is the linear one. On the other hand there is a huge class of Dehn functions $d(n)$ with growth at least $n^4$ (essentially all possible…

We consider the non-trivial zeros of the Riemann $\zeta$-function and two classes of $L$-functions; Dirichlet $L$-functions and those based on level one modular forms. We show that there are an infinite number of zeros on the critical line…

Denote $f(n):=\sum_{1\le k\le n} \tau(2^k-1)$, where $\tau$ is the number of divisors function. Motivated by a question of Paul Erd\H{o}s, we show that the sequence of ratios $f(2n)/f(n)$ is unbounded. We also present conditional results on…

Let $p$ be a fixed prime, and let $v(a)$ stand for the exponent of $p$ in the prime factorization of the integer $a$. Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Write $S$ for the maximum of…

We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the…

Let $f(t_1, \ldots, t_r, X)\in \mathbb{Z}[t_1, \ldots, t_r,X]$ be irreducible and let $a_1, \ldots, a_r\in \mathbb{Z} \smallsetminus \{0,\pm 1\}$. Under a necessary ramification assumption on $f$, and conditionally on the Generalized…

Let $f(n,\ell)$ be the maximum integer such that every set of $n$ points in the plane with at most $\ell$ collinear contains a subset of $f(n,\ell)$ points with no three collinear. First we prove that if $\ell \leq O(\sqrt{n})$ then…

We show that an arbitrary nilprogression can be approximated by a proper coset nilprogression in upper-triangular form. This can be thought of as a nilpotent version of the Freiman-Bilu result that a generalised arithmetic progression can…

This paper is devoted to studying the first-order variational analysis of non-convex and non-differentiable functions that may not be subdifferentially regular. To achieve this goal, we entirely rely on two concepts of directional…

Alon's combinatorial Nullstellensatz (Theorem 1.1 from \cite{Alon1}) is one of the most powerful algebraic tools in combinatorics, with a diverse array of applications. Let $\F$ be a field, $S_1,S_2,..., S_n$ be finite nonempty subsets of…