arXiv subjects

Pascal Jelinek

Publications and source records attributed to Pascal Jelinek.

5 recordsLinked to original sources

arXiv2026

Ratio of sum of digits functions in two bases

In 2019 La Bret\`eche, Stoll and Tennenbaum showed that the ratio of the sum of digits function $s_{q_1}(n)/s_{q_2}(n)$ of two multiplicatively independent bases $q_1$ and $q_2$ is dense in $\mathbb{Q}^+$. Recently Spiegelhofer proved that in the special case $s_2(n)/s_3(n)=1$ we have infinitely many solutions. Spiegelhofer extended this jointly with Drmota to show that the pair $(s_2(n),s_3(n))$ attains almost every value of $\mathbb{N}^2$ and hence, in particular, that every rational ratio is attained infinitely many times.\\ In this paper we show that, indeed, for any pair of multiplicatively independent bases $p$ and $q$, that the ratio attains every rational number infinitely many times. We also study this problem in the multiplicatively dependent case, hence giving a complete characterisation in the case of 2 bases.

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arXiv2026

Binomial coefficients coprime to 6

It is a well known result by Singmaster that any integer $d$ divides almost all binomial coefficients. The study of the structure and the size of the exceptional set has been of interest for many decades. In the case where $d$ is a prime power, these sets are well understood and their size is known. However, if $d$ has at least two distinct prime factors, no non-trivial bounds are known. In this paper, we will provide the first non trivial bound on the number of binomial coefficients coprime to 6.

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arXiv2025

Gowers norms for linearly recurrent numeration systems

Gowers norms have been a key component in the proofs of many breakthrough results in connection to the sum of digits function. Spiegelhofer has used them to show that the Thue-Morse sequence has level of distribution 1 and also that it is equidistributed along cubes. Recently Gowers norms have been used to study the sum of digits function of the Zeckendorf expansion of primes. In this paper we unify the treatments of Gowers norms and give Gowers norms type estimates for a large class of linearly recurrent numeration systems.

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arXiv2024

Binary-ternary collisions and the last significant digit of $n!$ in base 12

The third-named author recently proved [Israel J. of Math. 258 (2023), 475--502] that there are infinitely many \textit{collisions} of the base-2 and base-3 sum-of-digits functions. In other words, the equation \[ s_2(n)=s_3(n) \] admits infinitely many solutions in natural numbers. We refine this result and prove that every integer $a$ in $\{1, 2, \ldots, 11\}$ appears as the last nonzero digit of $n!$ in base $12$ infinitely often.

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arXiv2023

Square-free values of polynomials on average

The number of square-free integers in $x$ consecutive values of any polynomial $f$ is conjectured to be $c_fx$, where the constant $c_f$ depends only on the polynomial $f$. This has been proven for degrees less or equal to 3. Granville was able to show conditionally on the $abc$-conjecture that this conjecture is true for polynomials of arbitrarily large degrees. In 2013 Shparlinski proved that this conjecture holds on average over all polynomials of a fixed naive height, which was improved by Browning and Shparlinski in 2023. In this paper, we improve the dependence between $x$ and the height of the polynomial. We achieve this via adapting a method introduced in a 2022 paper by Browning, Sofos, and Ter\"av\"ainen on the Bateman-Horn conjecture, the polynomial Chowla conjecture, and the Hasse principle on average.

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