arXiv subjects

Youssef Lazar

Publications and source records attributed to Youssef Lazar.

7 recordsLinked to original sources

arXiv2025

A lattice point counting approach for the study of the number of self-avoiding walks on $\mathbb{Z}^{d}$

We reduce the problem of counting self-avoiding walks in the square lattice to a problem of counting the number of integral points in multidimensional domains. We obtain an asymptotic estimate of the number of self-avoiding walks of length $n$ in the square lattice. This new formalism gives a natural and unified setting in order to study the properties of the number of self-avoiding walks in the lattice $\mathbb{Z}^{d}$ of any dimension $d\geq 2$.

math.PR↗

arXiv2025

Simultaneous diophantine approximation for a restricted class of pairs of real numbers

We prove that the Littlewood conjecture is satisfied for a restricted class of pairs $(α,β)$ of badly approximable numbers. We use the localization of the roots of a cubic equation with coefficients depending on the diophantine properties for the considered pair $(α,β)$. The estimates of the roots rely on the properties of the denominators of the convergents of the continued fraction expansion of $α$ and $β$.

math.NT↗

arXiv2021

Explicit solutions to the Oppenheim conjecture for indefinite ternary diagonal forms

We prove the Oppenheim conjecture for indefinite ternary diagonal forms of the type $x^{2}+y^{2} -αz^{2}$ where $ α$ is an irrational number. Our method is explicit in the sense that we are able to construct a solution to the problem and we obtain an effective bound on the solution. The method is geometrical and is based on continued fractions.

math.NT↗

arXiv2020

Geometric considerations around the Littlewood conjecture

In this paper we adopt a geometric point of view regarding a famous conjecture due to Littlewood in diophantine approximation of real numbers. Following the spirit of the geometric theory of continued fractions, we give a sufficient condition for the conjecture to hold. An advantage of our method is that it is effective, in the sense that we know how to construct the eventual solution.

math.NT↗

arXiv2020

A remark on a conjecture of Erdős and Straus

The aim of this note is to show that given a positive integer $n \geq 5$, the positive integral solutions of the diophantine equation $4/n = 1/x + 1/y+1/z$ cannot have solution such that $x$ and $y$ are coprime with $xy < \sqrt{z/2}$. The proof uses the continued fraction expansion of $4/n$.

math.NT↗

arXiv2019

On the density of S-adic integers near some projective G-varieties

We provide some general conditions which ensure that a system of inequalities involving homogeneous polynomials with coefficients in a S-adic field has nontrivial S-integral solutions. The proofs are based on the strong approximation property for Zariski-dense subgroups and adelic geometry of numbers. We give two examples of applications for systems involving quadratic and linear forms.

math.NT↗

arXiv2016

Values of pairs involving one quadratic and one linear form at S-integral points

We prove the existence of S-integral solutions of simultaneous diophantine inequalities for pairs (Q,L) involving one quadratic form and one linear form satisfying some arithmetico-geometric conditions. The proof uses strong approximation in algebraic groups and Ratner's topological rigidity of unipotent actions on homogeneous spaces.

math.NT↗