Lukas Spiegelhofer
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View article: Decomposing the sum-of-digits correlation measure
Decomposing the sum-of-digits correlation measure Open
View article: Synchronizing automatic sequences along Piatetski-Shapiro sequences
Synchronizing automatic sequences along Piatetski-Shapiro sequences Open
The purpose of this paper is to study subsequences of synchronizing k -automatic sequences ( a ( n )) n ≥0 along Piatetski-Shapiro Sequences ⌊ n c ⌋ with c > 1. In particular we show that ( a (⌊ n c ⌋)) n ≥0 satisfies a prime number theore…
View article: The joint distribution of binary and ternary digits sums
The joint distribution of binary and ternary digits sums Open
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 s…
View article: Binary-ternary collisions and the last significant digit of $n!$ in base 12
Binary-ternary collisions and the last significant digit of $n!$ in base 12 Open
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 i…
View article: Decomposing the sum-of-digits correlation measure
Decomposing the sum-of-digits correlation measure Open
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 associat…
View article: Block occurrences in the binary expansion
Block occurrences in the binary expansion Open
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 $n$ suc…
View article: Thue--Morse along the sequence of cubes
Thue--Morse along the sequence of cubes Open
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…
View article: Synchronizing automatic sequences along Piatetski-Shapiro sequences
Synchronizing automatic sequences along Piatetski-Shapiro sequences Open
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 $a(\lfloor n^c \rfloor)$ satisfi…
View article: Möbius orthogonality of sequences with maximal entropy
Möbius orthogonality of sequences with maximal entropy Open
View article: GAPS IN THE THUE–MORSE WORD
GAPS IN THE THUE–MORSE WORD Open
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, or more precisely, the sequence of gaps between consecutiv…
View article: The binary digits of n+t
The binary digits of n+t Open
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)\g…
View article: Primes as sums of Fibonacci numbers
Primes as sums of Fibonacci numbers Open
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…
View article: Collisions of the binary and ternary sum-of-digits functions
Collisions of the binary and ternary sum-of-digits functions Open
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 binary sum of digits of $n$ equals its ternary sum of digits.
View article: Collisions of digit sums in bases 2 and 3
Collisions of digit sums in bases 2 and 3 Open
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 $n$.
View article: A lower bound for Cusick’s conjecture on the digits of <i>n</i> + <i>t</i>
A lower bound for Cusick’s conjecture on the digits of <i>n</i> + <i>t</i> Open
Let S be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t , define the asymptotic density $${c_t} = \mathop {\lim }\limits_{N \to \infty } {1…
View article: Sur la répartition jointe de la représentation d'Ostrowski dans les classes de résidue
Sur la répartition jointe de la représentation d'Ostrowski dans les classes de résidue Open
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 $α_2-$representati…
View article: The digits of n+t
The digits of n+t Open
The binary sum-of-digits function $s(n)$ counts the number of $\mathtt 1$s in the binary expansion of a nonnegative integer $n$. T.~W.~Cusick defined the asymptotic density \[ c_t=\lim_{N\rightarrow \infty} \frac 1N \lvert\{0\leq n 1/2$ fo…
View article: A lower bound for Cusick's conjecture on the binary digits of n+t
A lower bound for Cusick's conjecture on the binary digits of n+t Open
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} \f…
View article: A lower bound for Cusick's conjecture on the digits of n+t
A lower bound for Cusick's conjecture on the digits of n+t Open
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} \f…
View article: Approaching Cusick's conjecture on the sum-of-digits function
Approaching Cusick's conjecture on the sum-of-digits function Open
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 \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n1/2. \] We prove that for given $\vare…
View article: The Tu–Deng Conjecture holds Almost Surely
The Tu–Deng Conjecture holds Almost Surely Open
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$1\leqslant t<2^k-1$. Then\[\Bigl \…
View article: RANDOMNESS AND NON‐RANDOMNESS PROPERTIES OF PIATETSKI‐SHAPIRO SEQUENCES MODULO <i>m</i>
RANDOMNESS AND NON‐RANDOMNESS PROPERTIES OF PIATETSKI‐SHAPIRO SEQUENCES MODULO <i>m</i> Open
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 sequences. We prove that each block is an element of {0, 1}(k) of length k …
View article: Pseudorandomness of the Ostrowski sum-of-digits function
Pseudorandomness of the Ostrowski sum-of-digits function Open
For an irrational , we investigate the Ostrowski sum-of-digits function . For having bounded partial quotients and , we prove that the function , where , is pseudorandom in the following sense: for all the limit exists and we have
View article: Discrepancy Results for The Van Der Corput Sequence
Discrepancy Results for The Van Der Corput Sequence Open
Let d N = ND 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 ≤ log N/ 100. Moreover, we show that the summatory function of d N satisfies an exac…
View article: Divisibility of binomial coefficients by powers of two
Divisibility of binomial coefficients by powers of two Open
View article: Möbius orthogonality for the Zeckendorf sum-of-digits function
Möbius orthogonality for the Zeckendorf sum-of-digits function Open
We show that the (morphic) sequence is asymptotically orthogonal to all bounded multiplicative functions, where denotes the Zeckendorf sum-of-digits function. In particular we have , that is, this sequence satisfies the Sarnak conjecture.
View article: A Digit Reversal Property for Stern Polynomials
A Digit Reversal Property for Stern Polynomials Open
See the abstract in the attached pdf.
View article: A digit reversal property for an analogue of Stern's sequence
A digit reversal property for an analogue of Stern's sequence Open
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, $b_n=b_{n^R}$, where $n^R$ is obtained by reversing the bas…
View article: Normality of the Thue–Morse sequence along Piatetski-Shapiro sequences, II
Normality of the Thue–Morse sequence along Piatetski-Shapiro sequences, II Open
View article: The Maximal Order of Hyper-($b$-ary)-expansions
The Maximal Order of Hyper-($b$-ary)-expansions Open
Using methods developed by Coons and Tyler, we give a new proof of a recent result of Defant, by determining the maximal order of the number of hyper-($b$-ary)-expansions of a nonnegative integer $n$ for general integral bases $b\geqslant …