Searcharxiv⌕ Search

arXiv subjects

Dzmitry Badziahin

Publications and source records attributed to Dzmitry Badziahin.

At least 19 recordsLinked to original sources
arXiv2026

Positive Logarithmic Hausdorff Measures of Exceptional Sets for the $p$-adic and $t$-adic Littlewood Conjectures

We prove that if the exceptional set $E_p$ for the $p$-adic Littlewood conjecture is non-empty, then its logarithmic Hausdorff dimension is at least one. More precisely, whenever $E_p$ is non-empty, it has positive Hausdorff measure with respect to the gauge function $ h(r)=\frac{1}{\log(1/r)}. $ In particular, every non-empty $E_p$ has the cardinality of the continuum. We obtain stronger conclusions for the $t$-adic Littlewood conjecture over a finite field $\mathbb F_q$. For every prime power $q$, non-emptiness of the exceptional set $E_q^{(t)}$ implies that its $1/\log(1/r)$-Hausdorff measure is infinite. Moreover, when $q$ is odd, we refine the recent construction of Lai and Sprang~\cite{LaiSprang2026} and prove that \[ \mathcal H^{h_{A_q}}(E_q^{(t)})=\infty, \] where \[ h_{A_q}(r)=\frac{1}{(\log(1/r))^{A_q}}, \qquad A_q=\frac{q-1}{2}\log_2(q-1). \] In characteristic two, the corresponding conclusion with exponent one remains conditional on the existence of a counterexample.

math.NT↗
arXiv2026

On the $P(t)$-adic Littlewood Conjecture in Characteristics $\ell \equiv 3\pmod{4}$

Given a prime $p$, the $p$-adic Littlewood Conjecture stands as a well-known arithmetic variant of the celebrated Littlewood Conjecture in Diophantine Approximation. In the same way as the latter, it admits a natural function field analogue depending on the choice of an irreducible polynomial $P(t)$ with coefficients in a field $\mathbb{K}$. This analogue is referred to as the $P(t)$-adic Littlewood Conjecture ($P(t)$-LC for short). $P(t)$-LC is proved to fail for any choice of irreducible polynomial $P(t)$ over any ground field $\mathbb{K}$ with characteristic $ \ell \equiv 3\pmod{4}$. The counterexample refuting it is shown to present a local arithmetic obstruction emerging from the fact that -1 is not a quadratic residue modulo a prime $\ell\equiv 3\pmod{4}$. The theory developed elucidates and generalises all previous approaches towards refuting the conjecture. They were all based on the computer-assisted method initiated by Adiceam, Nesharim and Lunnon (2021) which has been able to establish that $P(t)$-LC fails in some small characteristics (essentially up to 11). This computer-assisted method is, however, unable to provide a general statement as it relies on ad hoc computer verifications which, provided they terminate, refute $P(t)$--LC in a given characteristic. This limitation is overcome by exhibiting an arithmetic obstruction to the validity of $P(t)$--LC in infinitely many characteristics. The existence of arithmetic obstructions within the context of $P(t)$--LC leaves the remaining case of odd characteristics $ \ell\equiv 1\pmod{4}$ dependent on their determination. This is shown to hold in an effective and explicit way.

math.NT↗
arXiv2025

Simultaneous Diophantine approximation to points on the Veronese curve

We compute the Hausdorff dimension of the set of simultaneously $q^{-λ}$-well approximable points on the Veronese curve in $\mathbb{R}^n$ for $λ$ between $\frac{1}{n}$ and $\frac{2}{2n-1}$. For $n=3$, the same result is given for a wider range of $λ$ between $\frac13$ and $\frac12$. We also provide a nontrivial upper bound for this Hausdorff dimension in the case $λ\le \frac{2}{n}$. In the course of the proof we establish that the number of cubic polynomials of height at most $H$ and non-zero discriminant at most $D$ is bounded from above by $c(ε) H^{2/3 + ε} D^{5/6}$.

math.NT↗
arXiv2023

On effective irrationality exponents of cubic irrationals

We provide an upper bound on the efficient irrationality exponents of cubic algebraics $x$ with the minimal polynomial $x^3 - tx^2 - a$. In particular, we show that it becomes non-trivial, i.e. better than the classical bound of Liouville in the case $|t| > 19.71 a^{4/3}$. Moreover, under the condition $|t| > 86.58 a^{4/3}$, we provide an explicit lower bound on the expression $||qx||$ for all large $q\in\mathbb{Z}$. These results are based on the recently discovered continued fractions of cubic irrationals and improve the currently best-known bounds of Wakabayashi.

math.NT↗
arXiv2021

On simultaneous rational approximation to a $p$-adic number and its integral powers, II

Let $p$ be a prime number. For a positive integer $n$ and a real number $ξ$, let $λ_n (ξ)$ denote the supremum of the real numbers $λ$ for which there are infinitely many integer tuples $(x_0, x_1, \ldots , x_n)$ such that $| x_0 ξ- x_1|_p, \ldots , | x_0 ξ^n - x_n|_p$ are all less than $X^{-λ- 1}$, where $X$ is the maximum of $|x_0|, |x_1|, \ldots , |x_n|$. We establish new results on the Hausdorff dimension of the set of real numbers $ξ$ for which $λ_n (ξ)$ is equal to (or greater than or equal to) a given value.

math.NT↗
arXiv2020

Schmidt games and Cantor winning sets

Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of study. We survey the definitions of the most common variants and connections between them. A new game called the Cantor game is invented and helps with presenting a unifying framework. We prove surprising new results such as the coincidence of absolute winning and $1$ Cantor winning in doubling metric spaces, and the fact that $1/2$ winning implies absolute winning for subsets of $\mathbb{R}$, and we suggest a prototypical example of a Cantor winning set to show the ubiquity of such sets in metric number theory and ergodic theory.

math.NT↗
arXiv2020

On t-adic Littlewood conjecture for generalised Thue-Morse functions

We consider a Laurent series defined by infinite products $g_u(t) = \prod_{n=0}^\infty (1 + ut^{-2^n})$, where $u\in \mathbb{F}$ is a parameter and $\mathbb{F}$ is a field. We show that for all $u\in\mathbb{Q}\setminus\{-1,0,1\}$ the series $g_u(t)$ does not satisfy the $t$-adic Littlewood conjecture. On the other hand, if $\mathbb{F}$ is finite then $g_u(t)\in \mathbb{F}((t^{-1}))$ is either a rational function or it satisfies the $t$-adic Littlewood conjecture.

math.NT↗
arXiv2019

An improved bound in Wirsing's problem

We improve the lower bound for the classical exponent of approximation $w_{n}^{\ast}(ξ)$ connected to Wirsing's famous problem of approximation to real numbers by algebraic numbers of degree at most $n$. Our bound exceeds $n/\sqrt{3}\approx 0.5773n$ and thus provides a reasonable qualitative improvement to previous bounds of order $n/2+O(1)$. We further establish new relations between several classical exponents of approximation.

math.NT↗
arXiv2018

On spectrum of irrationality exponents of Mahler numbers

We consider Mahler functions $f(z)$ which solve the functional equation $f(z) = \frac{A(z)}{B(z)} f(z^d)$ where $\frac{A(z)}{B(z)}\in \mathbb{Q}(z)$ and $d\ge 2$ is integer. We prove that for any integer $b$ with $|b|\ge 2$ either $f(b)$ is rational or its irrationality exponent is rational. We also compute the exact value of the irrationality exponent for $f(b)$ as soon as the continued fraction for the corresponding Mahler function is known. This improves the result of Bugeaud, Han, Wei and Yao where only an upper bound for the irrationality exponent was provided.

math.NT↗
arXiv2016

Continuant Diophantine equations

We investigate a family of Diophantine polynomial equations which involve continuant functions. In particular, given a polynomial $P(x)\in \mathbb{Z}[x]$ and $n\in \mathbb{N}$, we consider the equation $P(K_n(x_1,\ldots, x_n)) = K_{n+1}(x_0,\ldots,x_n)K_{n+1}(x_1,\ldots, x_{n+1})$. We show that with certain restrictions on $P(x)$ the set of its solutions has a rich structure. In particular, we provide several ways of generating new solutions from the existing ones. In the last section we discuss the relation between the solutions of the above Diophantine equation for arbitrary values of $n$ and factorisations $P(m) = d_1d_2$ for integers $m,d_1$ and $d_2$.

math.NT↗
arXiv2015

Cantor-winning sets and their applications

We introduce and develop a class of \textit{Cantor-winning} sets that share the same amenable properties as the classical winning sets associated to Schmidt's $(α,β)$-game: these include maximal Hausdorff dimension, invariance under countable intersections with other Cantor-winning sets and invariance under bi-Lipschitz homeomorphisms. It is then demonstrated that a wide variety of badly approximable sets appearing naturally in the theory of Diophantine approximation fit nicely into our framework. As applications of this phenomenon we answer several previously open questions, including some related to the Mixed Littlewood conjecture and the $\times2, \times3$ problem.

math.NT↗
arXiv2015

On generalized Thue-Morse functions and their values

This paper naturally extends and generalizes our previous work "Thue-Morse constant is not badly approximable", arXiv:1407.3182 [math.NT]. Here we consider the Laurent series $f_d(x) = \prod_{n=0}^\infty (1 - x^{-d^n})$, $d\in\mathbb{N}$, $d\geq 2$ which generalize the generating function $f_2(x)$ of the Thue-Morse number, and study their continued fraction expansion. In particular, we show that the convergents of $x^{-d+1}f_d(x)$ have quite a regular structure. We address as well the question whether the corresponding Mahler numbers $f_d(a)\in\mathbb{R}$, $a,d\in\mathbb{N}$, $a,d\geq 2$, are badly approximable.

math.NT↗
arXiv2015

An Unusual Continued Fraction

We consider the real number $σ$ with continued fraction expansion $[a_0, a_1, a_2,\ldots] = [1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,\ldots]$, where $a_i$ is the largest power of $2$ dividing $i+1$. We compute the irrationality measure of $σ^2$ and demonstrate that $σ^2$ (and $σ$) are both transcendental numbers. We also show that certain partial quotients of $σ^2$ grow doubly exponentially, thus confirming a conjecture of Hanna and Wilson.

math.NT↗
arXiv2015

An Inhomogeneous Jarník type theorem for planar curves

In metric Diophantine approximation there are two main types of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarník are fundamental in these settings. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximations on manifolds. In particular, both the Khintchine and Jarník type results have been established for planar curves except for only one case. In this paper, we prove an inhomogeneous Jarník type theorem for convergence on planar curves and in so doing complete the metric theory for both the homogeneous and inhomogeneous settings for approximation on planar curves.

math.NT↗
arXiv2015

On continued fraction expansion of potential counterexamples to $p$-adic Littlewood conjecture

The $p$-adic Littlewood conjecture (PLC) states that $\liminf_{q\to\infty} q\cdot |q|_p \cdot ||qx|| = 0$ for every prime $p$ and every real $x$. Let $w_{CF}(x)$ be an infinite word composed of the continued fraction expansion of $x$ and let $\mathrm{T}$ be the standard left shift map. Assuming that $x$ is a counterexample to PLC we get several restrictions on limit elements of the sequence $\{\mathrm{T}^n w_{CF}(x)\}_{n\in\mathbb{N}}$. As a consequence we show that for any such limit element $w$ we must have $\lim_{n\to\infty} P(w,n) - n = \infty$ where $P(w,n)$ is a word complexity of $w$. We also show that $w$ can not be among a certain collection of recursively constructed words.

math.NT↗
arXiv2014

Thue-Morse constant is not badly approximable

We prove that Thue-Morse constant $τ_{TM}=0.01101001..._2$ is not a badly approximable number. Moreover, we prove that $τ_{TM}(a)=0.01101001..._a$ is not badly approximable for every integer base $a\geq 2$ such that $a$ is not divisible by 15. At the same time we provide a precise formula for convergents of the Laurent series $\tilde{f}_{TM}(z) = z^{-1}\prod_{n=1}^\infty (1-z^{-2^n})$, thus developing further the research initiated by Alf van der Poorten and others.

math.NT↗
arXiv2013

Badly approximable points on planar curves and a problem of Davenport

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}, ||qy||^{1/j}} > c/q for all integers q >0. Here c = c(x,y) is a positive constant. Our main result implies that any finite intersection of such sets with C has full Hausdorff dimension. This provides a solution to a problem of Davenport dating back to the sixties.

math.NT↗