Searcharxiv⌕ Search

arXiv subjects

Dmitry Badziahin

Publications and source records attributed to Dmitry Badziahin.

12 recordsLinked to original sources
arXiv2026

Simultaneous Diophantine approximation on the three-dimensional Veronese curve: the complete Hausdorff dimension story

We determine the Hausdorff dimension of the intersection of the set $\mathcal W_3(\lambda)$ of simultaneously $\lambda$-well approximable points with the Veronese curve $\mathcal V_3 \subset \mathbb R^3$ for $\lambda \ge3/5$, thus completing the full range of $\lambda$ values. Precisely, we show that for $\lambda\ge 1/3$, $$ \dim\bigl(\mathcal W_3(\lambda)\cap\mathcal V_3\bigr)= \max\left\{\frac{2-2\lambda}{1+\lambda}, \frac{2}{3(1+\lambda)}\right\}. $$ To the best of the authors' knowledge, this makes $\mathcal V_3$ the first nondegenerate, non-planar curve with a completely determined Hausdorff dimension theory.

math.NT↗
arXiv2025

Distance between cubics and rationals

We investigate the following problem: what is the smallest possible distance between a cubic irrational $\xi$ and a rational number $p/q$ in terms of the height $H(\xi)$ and $q$? More precisely, we consider the set $D_{3,1}$ consisting of all pairs $(u,v)$ of positive real numbers such that $|\xi - p/q| > cH^{-u}(\xi)q^{-v}$ for all cubic irrationals $\xi$ and rationals $p/q$. First, we transform this problem into one about the root separation of cubic polynomials. Second, under the assumption of the famous abc-conjecture, we give an almost complete description of $D_{3,1}$. Namely, the points $(u,v)$ with $2\le v\le 3$ that lie in the interior of $D_{3,1}$ are characterised by the inequality $u> 10-3v$. Assuming only the weaker Hall conjecture, we also obtain nontrivial results about the shape of $D_{3,1}$, although these are not as strong as those derived from the abc-conjecture. Finally, we discuss an analogue of the set $D_{3,1}$ in function fields where we are able to give an almost complete description unconditionally.

math.NT↗
arXiv2025

Generating random factorisations of polynomial values

We construct algorithms that efficiently generate random factorisations of values $P(n)$ as products of two integers, where $P\in\mathbb{Z}[x]$ is a given quadratic or cubic monic polynomial. In other words, the algorithms produce random triples $(n,d_1,d_2)\in\mathbb{Z}^3$ that solve the Diophantine equation $P(n) = d_1d_2$. In the case where $P$ is cubic, such an algorithm allows the construction of an RSA key of $k$ bits that can be described using about $k/3$ bits of information. We also show how to construct a solution $(n,d_1,d_2)$ with the ratio $d_1/d_2$ arbitrarily close to any given positive real number. This proves that among all solutions $(n,d_1,d_2)$ of $P(n) = d_1d_2$ the ratios $d_1/d_2$ are dense in $(0,+\infty)$.

math.NT↗
arXiv2025

Simultaneous Diophantine approximation on the three dimensional Veronese curve

We compute the Hausdorff dimension of the set of simultaneously $λ$-well approximable points on the Veronese curve in $\RR^3$ for $1/3\le λ\le 3/5$. This range for $λ$ was predicted in the conjecture of Beresnevich and Yang from~\cite{ber_yan_2023}. To the best of the author's knowledge, this makes $\VVV_3$ the first nondegenerate curve in $\RR^n$, $n\ge 3$, to confirm the lower bound part of this conjecture.

math.NT↗
arXiv2025

Estimating lower limit in the $p$-adic Littlewood conjecture

We verify that $\liminf_{q\to\infty} q\cdot |q|_p\cdot ||qx||<ε$ for all real $x$, small primes $p$ and relatively small $ε$. This result supports the famous $p$-adic Littlewood conjecture which states that the above lower limit is equal to 0 for all $x\in\mathbb{R}$. In particular, the result is established for $p=2$ with $ε=1/25$. For $3\le p\le 29$, the upper bounds for $ε$ vary, but they are always at most $1/10$.

math.NT↗
arXiv2024

Continued fractions of cubic Laurent series

We construct continued fraction expansions for several families of the Laurent series in $\mathbb{Q}[[t^{-1}]]$. To the best of the author's knowledge, this is the first result of this kind since Gauss derived the continued fraction expansion for $(1+t)^r$, $r\in\mathbb{Q}$ in 1813. As an application, we apply an analogue of the hypergeometric method to one of those families and derive non-trivial lower bounds on the distance $|x - \frac{p}{q}|$ between one of the real roots of $3x^3 - 3tx^2-3ax+at$, $a,t\in\mathbb{Z}$ and any rational number, under relatively mild conditions on the parameters $a$ and $t$. We also show that every cubic irrational $x\in\mathbb{R}$ admits a (generalised) continued fraction expansion in a closed form that can be explicitly computed. Finally, we provide an infinite series of cubic irrationals $x$ that have arbitrarily (but finitely) many better-than-expected rational approximations. That is, they are such that for any $τ< 3+\frac{15\ln 2}{24}\approx 3.4332...$ the inequality $||qx|| < (H(x)^τ qe^{c\sqrt{\ln q}})^{-1}$ has many solutions in integer $q$.

math.NT↗
arXiv2021

Upper bounds for the uniform simultaneous Diophantine exponents

We give several upper bounds for the uniform simultaneous Diophantine exponent $\widehatλ_n(ξ)$ of a transcendental number $ξ\in\mathbb{R}$. The most important one relates $\widehatλ_n(ξ)$ and the ordinary simultaneous exponent $ω_k(ξ)$ in the case when $k$ is substantially smaller than $n$. In particular, in the generic case $ω_k(ξ)=k$ with a properly chosen $k$, the upper bound for $\widehatλ_n(ξ)$ becomes as small as $\frac{3}{2n} + O(n^{-2})$ which is substantially better than the best currently known unconditional bound of $\frac{2}{n} + O(n^{-2})$. We also improve an unconditional upper bound on $\widehatλ_n(ξ)$ for even values of $n$.

math.NT↗
arXiv2020

Badly approximable numbers, Kronecker's theorem, and diversity of Sturmian characteristic sequences

We give an optimal version of the classical ``three-gap theorem'' on the fractional parts of $n θ$, in the case where $θ$ is an irrational number that is badly approximable. As a consequence, we deduce a version of Kronecker's inhomogeneous approximation theorem in one dimension for badly approximable numbers. We apply these results to obtain an improved measure of sequence diversity for characteristic Sturmian sequences, where the slope is badly approximable.

math.NT↗
arXiv2019

On simultaneous rational approximation to a real number and its integral powers, II

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 arbitrarily large positive integers $q$ such that $|| q ξ||, || q ξ^2 ||, \ldots , ||q ξ^n||$ are all less than $q^{-λ}$. Here, $|| \cdot ||$ denotes the distance to the nearest integer. We establish new results on the Hausdorff dimension of the set of real numbers $ξ$ such that $λ_n (ξ)$ is equal (or greater than or equal) to a given value.

math.NT↗
arXiv2018

Continued fractions of certain Mahler functions

We investigate the continued fraction expansion of the infinite products $g(x) = x^{-1}\prod_{t=0}^\infty P(x^{-d^t})$ where polynomials $P(x)$ satisfy $P(0)=1$ and $°(P) 1$ such that $g(b)\neq0$ the irrationality exponent of $g(b)$ equals two. In the case $d=3$ we provide a partial analogue of the last result with several collections of polynomials $P(x)$ giving the irrationality exponent of $g(b)$ strictly bigger than two.

math.NT↗
arXiv2015

On the complexity of a putative counterexample to the $p$-adic Littlewood conjecture

Let $|| \cdot ||$ denote the distance to the nearest integer and, for a prime number $p$, let $| \cdot |_p$ denote the $p$-adic absolute value. In 2004, de Mathan and Teulié asked whether $\inf_{q \ge 1} \, q \cdot || q α|| \cdot | q |_p = 0$ holds for every badly approximable real number $α$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number $α$ grows too rapidly or too slowly, then their conjecture is true for the pair $(α, p)$ with $p$ an arbitrary prime.

math.NT↗