arXiv subjects

Lukas Spiegelhofer

Publications and source records attributed to Lukas Spiegelhofer.

At least 19 recordsLinked to original sources

arXiv2025

Decomposing the sum-of-digits correlation measure

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 \[s(n+t)-s(n),\] for all $n\ge0$, we consider the associated characteristic function $γ_t$. Our main theorem is a structural result on the decomposition of $γ_t$ into a sum of \emph{components}. We also study in detail the case that $t$ contains at most two blocks of consecutive $1$s. The results in this paper are motivated by \emph{Cusick's conjecture} on the sum-of-digits function. This conjecture is concerned with the \emph{central tendency} of the corresponding probability distributions, and is still unsolved.

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arXiv2025

The joint distribution of binary and ternary digits sums

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 represent a nonnegative integer as their sum. A result of the second author states that there are infinitely many \emph{collisions} of $s_2$ and $s_3$, that is, positive integers $n$ such that \[s_2(n)=s_3(n).\] This resolved a long-standing folklore conjecture. In the present paper, we prove a strong generalization of this statement, stating that $(s_2(n),s_3(n))$ attains almost all values in $\mathbb N^2$, in the sense of asymptotic density. In particular, this yields \emph{generalized collisions}: for any pair $(a,b)$ of positive integers, the equation \[as_2(n)=bs_3(n)\] admits infinitely many solutions in $n$.

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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

Thue--Morse along the sequence of cubes

The Thue--Morse sequence $t=01101001\cdots$ is an automatic sequence over the alphabet $\{0,1\}$. It can be defined as the binary sum-of-digits function $s:\mathbb N\rightarrow\mathbb N$, reduced modulo $2$, or by using the substitution $0\mapsto 01$, $1\mapsto 10$. We prove that the asymptotic density of the set of natural numbers $n$ satisfying $t(n^3)=0$ equals $1/2$. Comparable results, featuring asymptotic equivalence along a polynomial as in our theorem, were previously only known for the linear case [A. O. Gelfond, Acta Arith. 13 (1967/68), 259--265], and for the sequence of squares. The main theorem in [C. Mauduit and J. Rivat, Acta Math. 203 (2009), no. 1, 107--148] was the first such result for the sequence of squares. Concerning the sum-of-digits function along polynomials $p$ of degree at least three, previous results were restricted either to lower bounds (such as for the numbers $\#\{n<N:t(p(n))=0\}$), or to sum-of-digits functions in ``sufficiently large bases''. By proving an asymptotic equivalence for the case of the Thue--Morse sequence, and a cubic polynomial, we move one step closer to the solution of the third Gelfond problem on the sum-of-digits function (1967/1968), op. cit.

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arXiv2023

Block occurrences in the binary expansion

The binary sum-of-digits function $\mathsf{s}$ returns the number of ones in the binary expansion of a nonnegative integer. Cusick's Hamming weight conjecture states that, for all integers $t\geq 0$, the set of nonnegative integers $n$ such that $\mathsf{s}(n+t)\geq \mathsf{s}(n)$ has asymptotic density strictly larger than $1/2$. We are concerned with the block-additive function $\mathsf{r}$ returning the number of (overlapping) occurrences of the block $\mathtt{11}$ in the binary expansion of $n$. The main result of this paper is a central limit-type theorem for the difference $\mathsf{r}(n+t)-\mathsf{r}(n)$: the corresponding probability function is uniformly close to a Gaussian, where the uniform error tends to $0$ as the number of blocks of ones in the binary expansion of $t$ tends to $\infty$.

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arXiv2022

Synchronizing automatic sequences along Piatetski-Shapiro sequences

The purpose of this paper is to study subsequences of synchronizing $k$-automatic sequences $a(n)$ along Piatetski-Shapiro sequences $\lfloor n^c \rfloor$ with non-integer $c>1$. In particular, we show that $a(\lfloor n^c \rfloor)$ satisfies a prime number theorem of the form $\sum_{n\le x} Λ(n)a(\lfloor n^c \rfloor) \sim C\, x$, and, furthermore, that it is deterministic for $c \in \mathbb R\setminus \mathbb Z$. As an interesting additional result, we show that the sequence $\lfloor n^c\rfloor \bmod m$ has polynomial subword complexity.

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arXiv2022

Primes as sums of Fibonacci numbers

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 $k$ there exists a prime number that can be represented as the sum of $k$ different and non-consecutive Fibonacci numbers. This property is closely related to, and based on, a prime number theorem for certain morphic sequences. The proof of such a prime number theorem, combined with a corresponding local result, is the central contribution of this paper, from which we derive the result stated in the beginning. Problems of this type have been discussed intensively in the context of the base-$q$ expansion. The Gelfond problems (1968/1969), and the Sarnak conjecture, were the driving forces of this development. Mauduit and Rivat resolved the question on the sum of digits of prime numbers (2010) and the sum of digits of squares (2009), thus leaving open only part of the third Gelfond problem. Later the second author (2017) proved Sarnak's conjecture for all automatic sequences, which are based on the q-ary expansion of integers, and which generalize the sum-of-digits function in base $q$ considerably. In order to obtain corresponding results for Fibonacci numbers, we have to extend Mauduit and Rivat's method considerably. In fact, we are departing significantly from this method, proving the statement that $\exp(2πi \vartheta\mathsf z(n))$ has \emph{level of distribution} $1$ (here $\mathsf z(n)$ is the number of Fibonacci numbers needed to write $n$ as their sum). This latter result forms an essential part of our treatment of the occurring sums of type $\textrm I$ and $\textrm{II}$ and uses Gowers norms related to $\mathsf z(n)$ as a central technical tool. The appearance of Gowers norms in our method is intimately tied to the iterated application of a new generalization of van der Corput's inequality.

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arXiv2022

The binary digits of n+t

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 asymptotic density $c_t$ of integers $n\geq 0$ such that \[s(n+t)\geq s(n).\] In 2011, he conjectured that $c_t>1/2$ for all $t$ -- the binary sum of digits should, more often than not, weakly increase when a constant is added. In this paper, we prove that there exists an explicit constant $M_0$ such that indeed $c_t>1/2$ if the binary expansion of $t$ contains at least $M_0$ maximal blocks of contiguous ones, leaving open only the "initial cases" -- few maximal blocks of ones -- of this conjecture. Moreover, we sharpen a result by Emme and Hubert (2019), proving that the difference $s(n+t)-s(n)$ behaves according to a Gaussian distribution, up to an error tending to $0$ as the number of maximal blocks of ones in the binary expansion of $t$ grows.

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arXiv2022

Collisions of digit sums in bases 2 and 3

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 of $n$ equals the sum of the ternary digits of $n$.

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arXiv2021

Gaps in the Thue--Morse word

The Thue--Morse sequence is a prototypical automatic sequence found in diverse areas of mathematics, and in computer science. We study occurrences of factors $w$ within this sequence, more precisely, the sequence of gaps between consecutive occurrences. This gap sequence is morphic; we prove that it is not automatic as soon as the length of $w$ is at least two, thereby answering a question by J.~Shallit in the affirmative. We give an explicit method to compute the \emph{discrepancy} of the number of occurrences of the block $\mathtt{01}$ in the Thue--Morse sequence. We prove that the sequence of discrepancies is the sequence of output sums of a certain base-$2$ transducer.

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arXiv2020

Sur la répartition jointe de la représentation d'Ostrowski dans les classes de résidue

For two distinct integers $m_1,m_2\ge2$, we set $α_1=[0;\overline{1,m_1}]$ and $α_2=[0;\overline{1,m_2}]$ and we denote by $S_{α_1}(n)$ and $S_{α_2}(n)$ respectively the sum of digits functions in the Ostrowski $α_1$ and $α_2-$representations of $n$. Let $b_1,b_2 $ be positive integers satisfying $(b_1,m_1)=1$ and $(b_2,m_2)=1$, we obtain an estimation with an error term $O(N^{1-δ})$ for the cardinal of the following set $$\Big\{ 0\leq n<N;\ S_{α_1}(n)\equiv a_1\pmod{b_1},\ S_{α_2}(n)\equiv a_2\pmod{b_2}\Big\},$$ for all integers $a_1$ and $a_2.$ Our result should be compared to that of Bésineau and Kim who treated the case of the $q-$representations in different bases (that are coprimes).

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arXiv2020

Möbius orthogonality of sequences with maximal entropy

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 maximal entropy and satisfying this property.

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arXiv2019

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

Let $s$ be the sum-of-digits function in base $2$, which returns the number of $\mathtt 1$s in the base-2 expansion of a nonnegative integer. For a nonnegative integer $t$, define the asymptotic density \[ c_t=\lim_{N\rightarrow \infty} \frac 1N\bigl\lvert\{0\leq n 1/2$. We have the elementary bound $0 1/2-\varepsilon$ as soon as $t$ contains sufficiently many blocks of $\mathtt 1$s in its binary expansion. In the proof, we provide estimates for the moments of an associated probability distribution; this extends the study initiated by Emme and Prikhod'ko (2017) and pursued by Emme and Hubert (2018).

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arXiv2019

The sum-of-digits function on arithmetic progressions

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 and show that --- up to a shift --- the set of $m$-tuples of integers that appear as an arithmetic subsequence of $s_2$ has full complexity.

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arXiv2019

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

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 \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n 1/2. \] We prove that for given $\varepsilon>0$ we have \[ c_t+c_{t'}>1-\varepsilon \] if the binary expansion of $t$ contains enough blocks of consecutive $\mathtt 1$s (depending on $\varepsilon$), where $t'=3\cdot 2^λ-t$ and $λ$ is chosen such that $2^λ\leq t<2^{λ+1}$.

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arXiv2018

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

We study Piatetski-Shapiro sequences $(\lfloor n^c\rfloor)_n$ modulo m, for non-integer $c >1$ and positive $m$, and we are particularly interested in subword occurrences in those sequences. We prove that each block $\in\{0,1\}^k$ of length $k < c + 1$ occurs as a subword with the frequency $2^{-k}$, while there are always blocks that do not occur. In particular, those sequences are not normal. For $1<c<2$, we estimate the number of subwords from above and below, yielding the fact that our sequences are deterministic and not morphic. Finally, using the Daboussi-Kátai criterion, we prove that the sequence $\lfloor n^c\rfloor$ modulo m is asymptotically orthogonal to multiplicative functions bounded by $1$ and with mean value $0$.

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arXiv2018

The Tu--Deng Conjecture holds almost surely

The Tu--Deng Conjecture is concerned with the sum of digits $w(n)$ of $n$ in base~$2$ (the Hamming weight of the binary expansion of $n$) and states the following: assume that $k$ is a positive integer and $1\leq t<2^k-1$. Then \[\Bigl \lvert\Bigl\{(a,b)\in\bigl\{0,\ldots,2^k-2\bigr\}^2:a+b\equiv t\bmod 2^k-1, w(a)+w(b)<k\Bigr\}\Bigr \rvert\leq 2^{k-1}.\] We prove that the Tu--Deng Conjecture holds almost surely in the following sense: the proportion of $t\in[1,2^k-2]$ such that the above inequality holds approaches $1$ as $k\rightarrow\infty$. Moreover, we prove that the Tu--Deng Conjecture implies a conjecture due to T.~W.~Cusick concerning the sum of digits of $n$ and $n+t$.

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arXiv2018

Statistical distribution of the Stern sequence

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 analytic properties of a transfer operator.

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