Lukas Spiegelhofer

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20152025

most citedNormality of the Thue--Morse sequence along Piatetski-Shapiro sequences, II

21 citations · 60 across the 22 of their papers we have counts for

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26 papers · 1 filter

math.NT2025

The joint distribution of binary and ternary digits sums

Michael Drmota, Lukas Spiegelhofer

We consider the sum-of-digits functions $s_{2}$ and $s_{3}$ in bases $2$ and $3$ . These functions just return the minimal numbers of powers of two (resp. three) needed in order to repr…

math.NT2024

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

Jean-Marc Deshouillers, Pascal Jelinek, Lukas Spiegelhofer

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…

math.NT2024

Decomposing the sum-of-digits correlation measure

Bartosz Sobolewski, Lukas Spiegelhofer

Let $s (n)$ denote the number of ones in the binary expansion of the nonnegative integer $n$ . How does $s$ behave under addition of a constant $t$ ? In order to study the differences…

math.NT2023★ 1 cited

Thue--Morse along the sequence of cubes

Lukas Spiegelhofer

The Thue--Morse sequence $t = 01101001 \dots$ is an automatic sequence over the alphabet ${0, 1}$ . It can be defined as the binary sum-of-digits function $s:\mathbb N\rightarrow\mat…

math.NT2023

Block occurrences in the binary expansion

Bartosz Sobolewski, Lukas Spiegelhofer

The binary sum-of-digits function $s$ returns the number of ones in the binary expansion of a nonnegative integer. Cusick's Hamming weight conjecture states that, for all…

math.NT2022★ 2 cited

Synchronizing automatic sequences along Piatetski-Shapiro sequences

Jean-Marc Deshouillers, Michael Drmota, Clemens Müllner +2

The purpose of this paper is to study subsequences of synchronizing $k$ -automatic sequences $a (n)$ along Piatetski-Shapiro sequences $⌊ n^{c} ⌋$ with non-integer $c > 1$ . I…

math.NT2021★ 1 cited

Primes as sums of Fibonacci numbers

Michael Drmota, Clemens Müllner, Lukas Spiegelhofer

The purpose of this paper is to discuss the relationship between prime numbers and sums of Fibonacci numbers. One of our main results says that for every sufficiently large integer…

math.NT2021

Collisions of digit sums in bases 2 and 3

Lukas Spiegelhofer

We prove a folklore conjecture concerning the sum-of-digits functions in bases two and three: there are infinitely many positive integers $n$ such that the sum of the binary digits…

math.NT2020

The binary digits of n+t

Lukas Spiegelhofer, Michael Wallner

The binary sum-of-digits function $s$ counts the number of ones in the binary expansion of a nonnegative integer. For any nonnegative integer $t$ , T.~W.~Cusick defined the asymptot…

math.NT2020

Möbius orthogonality of sequences with maximal entropy

Michael Drmota, Christian Mauduit, Joël Rivat +1

We prove that strongly $b$ -multiplicative functions of modulus $1$ along squares are asymptotically orthogonal to the Möbius function. This provides examples of sequences having ma…

math.NT2019★ 2 cited

A lower bound for Cusick's conjecture on the digits of n+t

Lukas Spiegelhofer

Let $s$ be the sum-of-digits function in base $2$ , which returns the number of $1$ s in the base-2 expansion of a nonnegative integer. For a nonnegative integer $t$ , define…

math.NT2019

The sum-of-digits function on arithmetic progressions

Lukas Spiegelhofer, Thomas Stoll

Let $s_{2}$ be the sum-of-digits function in base $2$ , which returns the number of non-zero binary digits of a nonnegative integer $n$ . We study $s_{2}$ alon g arithmetic subsequences…

math.NT2019★ 3 cited

Approaching Cusick's conjecture on the sum-of-digits function

Lukas Spiegelhofer

Cusick's conjecture on the binary sum of digits $s (n)$ of a nonnegative integer $n$ states the following: for all nonnegative integers $t$ we have \[ c_t=\lim_{N\rightarrow\infty}\…

math.NT2018★ 5 cited

Randomness and non-randomness properties of Piatetski-Shapiro sequences modulo m

Jean-Marc Deshouillers, Michael Drmota, Clemens Müllner +1

We study Piatetski-Shapiro sequences $(⌊ n^{c} ⌋)_{n}$ modulo m, for non-integer $c > 1$ and positive $m$ , and we are particularly interested in subword occurrences in those…

math.NT2018★ 9 cited

The level of distribution of the Thue--Morse sequence

Lukas Spiegelhofer

The level of distribution of a complex valued sequence $b$ measures "how well $b$ behaves" on arithmetic progressions $n d + a$ . Determining whether $θ$ is a level of distribution for…

math.NT2017★ 1 cited

Divisibility of binomial coefficients by powers of two

Lukas Spiegelhofer, Michael Wallner

For nonnegative integers $j$ and $n$ let $Θ (j, n)$ be the number of entries in the $n$ -th row of Pascal's triangle that are not divisible by $2^{j + 1}$ . In this paper we prove that t…

math.NT2017

Discrepancy results for the Van der Corput sequence

Lukas Spiegelhofer

Let $d_{N} = N D_{N} (ω)$ be the discrepancy of the Van der Corput sequence in base $2$ . We improve on the known bounds for the number of indices $N$ such that $d_{N} \leq lo g N /100$ . Moreov…

math.NT2017★ 1 cited

A digit reversal property for an analogue of Stern's sequence

Lukas Spiegelhofer

We consider a variant of Stern's diatomic sequence, studied recently by Northshield. We prove that this sequence $b$ is invariant under \emph{digit reversal} in base $3$ , that is,…

math.NT2017★ 9 cited

Normality of the Thue--Morse sequence along Piatetski-Shapiro sequences

Lukas Spiegelhofer

We prove that for $1 < c < 4/3$ the subsequence of the Thue--Morse sequence $t$ indexed by $⌊ n^{c} ⌋$ defines a normal sequence, that is, each finite sequence $(\vare…

math.NT2017

Piatetski-Shapiro sequences via Beatty sequences

Lukas Spiegelhofer

Integer sequences of the form $⌊ n^{c} ⌋$ , where $1 < c < 2$ , can be locally approximated by sequences of the form $⌊ n α + β ⌋$ in a very good way. Following this app…

math.NT2017

Möbius orthogonality for the Zeckendorf sum-of-digits function

Michael Drmota, Clemens Müllner, Lukas Spiegelhofer

We show that the (morphic) sequence $(- 1)^{s_{φ} (n)}$ is asymptotically orthogonal to all bounded multiplicative functions, where $s_{φ}$ denotes the Zeckendorf sum-of-digits function.…

math.NT2017★ 1 cited

Statistical distribution of the Stern sequence

Sandro Bettin, Sary Drappeau, Lukas Spiegelhofer

We prove that the Stern diatomic sequence is asymptotically distributed according to a normal law, on a logarithmic scale. This is obtained by studying complex moments, and the ana…

math.NT2016★ 2 cited

Pseudorandomness of the Ostrowski sum-of-digits function

Lukas Spiegelhofer

For an irrational $α \in (0, 1)$ , we investigate the Ostrowski sum-of-digits function $σ_{α}$ . For $α$ having bounded partial quotients and $ϑ \in R ∖ Z$ , we…

math.NT2016

A digit reversal property for Stern polynomials

Lukas Spiegelhofer

We consider the following polynomial generalization of Stern's diatomic series: let $s_{1} (x, y) = 1$ , and for $n \geq 1$ set $s_{2 n} (x, y) = s_{n} (x, y)$ and $s_{2n+1}(x,y)=x\,s_n(x,y)+y\,s_{…

The joint distribution of binary and ternary digits sums

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

Decomposing the sum-of-digits correlation measure

Thue--Morse along the sequence of cubes

Block occurrences in the binary expansion

Synchronizing automatic sequences along Piatetski-Shapiro sequences

Primes as sums of Fibonacci numbers

Collisions of digit sums in bases 2 and 3

The binary digits of n+t

Möbius orthogonality of sequences with maximal entropy

A lower bound for Cusick's conjecture on the digits of n+t

The sum-of-digits function on arithmetic progressions

Approaching Cusick's conjecture on the sum-of-digits function

Randomness and non-randomness properties of Piatetski-Shapiro sequences modulo m

The level of distribution of the Thue--Morse sequence

Divisibility of binomial coefficients by powers of two

Discrepancy results for the Van der Corput sequence

A digit reversal property for an analogue of Stern's sequence

Normality of the Thue--Morse sequence along Piatetski-Shapiro sequences

Piatetski-Shapiro sequences via Beatty sequences

Möbius orthogonality for the Zeckendorf sum-of-digits function

Statistical distribution of the Stern sequence

Pseudorandomness of the Ostrowski sum-of-digits function

A digit reversal property for Stern polynomials

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