Related papers: On the Littlewood conjecture in simultaneous Dioph…

We show how a recent result by Aka and Shapira on the evolution of continued fractions in a fixed quadratic field implies the classic result of de Mathan and Teulli\'e on the Mixed Littlewood

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…

The Littlewood Conjecture states that liminf_{q\to \infty} q . ||qx|| . ||qy|| = 0 for all pairs (x,y) of real numbers. We show that with the additional factor of log q . loglog q the statement is false. Indeed, our main result implies that…

We prove that for every irrational number $\alpha$, real number $\beta$, real number $c$ satisfying $1<c<9/8$ and positive real number $\theta$ satisfying $\theta<(9/c-8)/10$, there exist infinitely many primes of the form…

Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…

Let $\alpha$ and $\beta$ be real numbers such that $1$, $\alpha$ and $\beta$ are linearly independent over $\mathbb{Q}$. A classical result of Dirichlet asserts that there are infinitely many triples of integers $(x_0,x_1,x_2)$ such that…

Given any polynomial with real coefficients, the existence of a real quadratic polynomial factor is proven using only basic real analysis. The aim is to provide an approachable proof to anybody who is familiar with the least upper bound…

Zaremba's conjecture concerns a formation of continued fraction expansions for rational numbers with partial quotient bounded by an absolute constant. We present asymptotic estimates for the size of $\epsilon$-thickening of certain fractal…

We prove upper and lower bounds for certain sums of products of fractional parts by using majoring and minorizing functions from Fourier analysis. In special cases the upper bounds are sharp if there exist counterexamples to the Littlewood…

We study quadratic approximations for two families of hyperquadratic continued fractions in the field of Laurent series over a finite field. As the first application, we give the answer to a question of the second author concerning…

We prove a parabolic version of the Littlewood-Paley inequality for the fractional Laplacian $(-\Delta)^{\alpha/2}$, where $\alpha\in (0,2)$.

A {\it two-dimensional continued fraction expansion} is a map $\mu$ assigning to every $x \in\mathbb R^2\setminus\mathbb Q^2$ a sequence $\mu(x)=T_0,T_1,\dots$ of triangles $T_n$ with vertices $x_{ni}=(p_{ni}/d_{ni},q_{ni}/d_{ni})\in\mathbb…

Let $\beta > 1$ be a real number and $x \in [0,1)$ be an irrational number. Denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $\beta$-expansion of $x$…

The classical Khintchine and Jarn\'ik theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real…

We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the…

In this paper, we improve the results in the author's previous paper \cite{Usu22}, which deals with the quantitative problem on Littlewood's conjecture. We show that, for any $0<\gamma<1$, any $(\alpha,\beta)\in\mathbb{R}^2$ except on a set…

In this paper we introduce the concept of an infinite loop mod $n$ and discuss the properties that these objects have. In particular, we show that a real number $\alpha$ is a counterexample to the $p$-adic Littlewood Conjecture if and only…

We prove a strong analogue of Liouville's Theorem in Diophantine approximation for points on arbitrary algebraic varieties. We use this theorem to prove a conjecture of the first author for cubic surfaces in $\P^3$.

The present paper is in a sense a continuation of \cite{PLS}, it relies on the notation and some results. The problem tackled in both papers is the nature of the continued fraction expansion of $\sqrt[3]{2}$: are the partial quotients…

Legendre's theorem states that every irreducible fraction $\frac{p}{q}$ which satisfies the inequality $\left |\alpha-\frac{p}{q} \right | < \frac{1}{2q^2}$ is convergent to $\alpha$. Later Barbolosi and Jager improved this theorem. In this…