Related papers: On multiplicatively badly approximable numbers

We show that there exist absolute constants $\Delta > \delta > 0$ such that, for all $n \geqslant 2$, there exists a polynomial $P$ of degree $n$, with $\pm 1$ coefficients, such that $$\delta\sqrt{n} \leqslant |P(z)| \leqslant…

We derive by lattice theory a universal quantum certainty relation for arbitrary $M$ observables in $N$-dimensional system, which provides a state-independent maximum lower bound on the direct-sum of the probability vectors in terms of…

In 2016, Fei \cite{fei2016application} established a bound on the Siegel zeros for real primitive Dirichlet characters modulo $q$, assuming the weak Hardy-Littlewood conjecture. Building on Fei's work, Jia \cite{jia2022conditional}…

The theory of uniform approximation of real numbers motivates the study of products of consecutive partial quotients in regular continued fractions. For any non-decreasing positive function $\varphi:\mathbb{N}\to [2,\infty)$, we determine…

We show that $0,1$-polynomials of high degree and few terms are irreducible with high probability. Formally, let $k\in\mathbb{N}$ and $F(x)=1+\sum_{i=1}^kx^{n_i}$, where $ 0<n_1<\cdots<n_k\leq N. $ Then we show that…

The Hardy--Littlewood inequalities for multilinear forms on sequence spaces state that for all positive integers $m,n\geq2$ and all $m$-linear forms $T:\ell_{p_{1}}^{n}\times\cdots\times\ell_{p_{m}}^{n}\rightarrow\mathbb{K}$…

We give a simple proof of a recent result by J. Schleischitz dealing with a counterexample to the uniform Littlewood conjecture. Our construction is based on simple properties of Fibonacci numbers.

We prove the following results: let x,y be (n,n) complex matrices such that x,y,xy have no eigenvalue in ]-infinity,0] and log(xy)=log(x)+log(y). If n=2, or if n>2 and x,y are simultaneously triangularizable, then x,y commute. In both cases…

Let $M_d$ be the centered Hardy-Littlewood maximal function associated to cubes in $\mathbb{R}^d$ with Lebesgue measure, and let $c_d$ denote the lowest constant appearing in the weak type (1,1) inequality satisfied by $M_d$. We show that…

By means of principal isotopes lH(a,b) of the algebra lH [Ra 99] we give an exhaustive and not repetitive description of all 4-dimensional absolute-valued algebras satisfying (x^p, x^q, x^r) = 0 for fixed integers p, q, r \in\{1,2\}. For…

We prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.

Let C be two times continuously differentiable curve in R^2 with at least one point at which the curvature is non-zero. For any i,j > 0 with i+j =1, let Bad(i,j) denote the set of points (x,y) in R^2 for which max {||qx ||^{1/i},…

We show that if $A$ is a finite set of non-negative integers then the number of zeros of the function \[ f_A(\theta) = \sum_{a \in A} \cos(a\theta), \] in $[0,2\pi]$, is at least $(\log \log \log |A|)^{1/2-\varepsilon}$. This gives the…

In Alm-Hirsch-Maddux (2016), relation algebras $\mathfrak{L}(q,n)$ were defined that generalize Roger Lyndon's relation algebras from projective lines, so that $\mathfrak{L}(q,0)$ is a Lyndon algebra. In that paper, it was shown that if…

Let $M_{d,n}(q)$ denote the number of monic irreducible polynomials in $\mathbb{F}_q[x_1, x_2, \ldots , x_n]$ of degree $d$. We show that for a fixed degree $d$, the sequence $M_{d,n}(q)$ converges $q$-adically to an explicitly determined…

Let $Q(x,y,z)$ be an integral quadratic form with determinant coprime to some modulus $q$. We show that $q\mid Q$ for some non-zero integer vector $(x,y,z)$ of length $O(q^{5/8+\varepsilon})$, for any fixed $\varepsilon>0$. Without the…

The Jarn\'ik-Besicovitch theorem is a fundamental result in metric number theory which concerns the Hausdorff dimension for certain limsup sets. We discuss the analogous problem for liminf sets. Consider an infinite sequence of positive…

Given a strictly increasing sequence of positive real numbers tending to infinity $(q_{n})_{n=1}^{\infty}$, and an arbitrary sequence of real numbers $(r_{n})_{n=1}^{\infty}.$ We study the set of $\alpha\in(1,\infty)$ for which…

The first part of this paper deals with unipotent and reductive groups over finite fields with $q$ elements in which either $q$ goes to infinity or $G=GL_n(q)$ and $n$ goes to infinity. The second part of the paper deals with the symmetric…

We show that for every mean zero log-concave real random variable $X$ one has $\|X\|_p \leq \frac{p}{q} \|X\|_q$ for $p \geq q \geq 1$, going beyond the well-known case of symmetric random variables. We also prove that in the class of…