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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 $n$., Comment: 24 pages

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 $\sum_{n\le x} \Lambda(n)a(\lfloor n^c \rfloor) \sim C\, x$, 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., Comment: 32 pages

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 $\exp(2\pi i \vartheta\mathsf z(n))$ 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., Comment: 135 pages, 1 figure

In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic sequence along squares $(n^2)_{n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, we give for these subsequences a criterion to decide whether the densities exist, in which case they are also computable. In particular in the prime case these densities are all rational. We also deduce from a recent result of the third author and Lema\'nczyk that all subshifts generated by automatic sequences are orthogonal to any bounded multiplicative aperiodic function., Comment: 35 pages. We added an Appendix concerning upper densities of subsequences of automatic sequences

In 1972 Delange observed in analogy of the classical Erd\H os-Wintner theorem that $q$-additive functions $f(n)$ has a distribution function if and only if the two series $\sum f(d q^j)$, $\sum f(d q^j)^2$ converge. The purpose of this paper is to provide quantitative versions of this theorem as well as generalizations to other kinds of digital expansions. In addition to the $q$-ary and Cantor case we focus on the Zeckendorf expansion that is based on the Fibonacci sequence, where we provide a sufficient and necessary condition for the existence of a distribution function, namely that the two series $\sum f(F_j)$, $\sum f(F_j)^2$ converge (previously only a sufficient condition was known)., Comment: 26 pages

We prove that strongly $b$-multiplicative functions of modulus $1$ along squares are asymptotically orthogonal to the M\"obius function. This provides examples of sequences having maximal entropy and satisfying this property., Comment: 12 pages

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

We prove a central limit theorem for the joint distribution of $s_q(A_jn)$, $1\le j \le d$, where $s_q$ denotes the sum-of-digits function in base~$q$ and the $A_j$'s are positive integers relatively prime to $q$. We do this in fact within the framework of quasi-additive functions. As application, we show that most elements of "Catalan-like" sequences - by which we mean integer sequences defined by products/quotients of factorials - are divisible by any given positive integer., Comment: AmS-LaTeX, 11 pages; minor corrections and improvements

We show that the (morphic) sequence $(-1)^{s_\varphi(n)}$ is asymptotically orthogonal to all bounded multiplicative functions, where $s_\varphi$ denotes the Zeckendorf sum-of-digits function. In particular we have $\sum_{n

For a nonnegative integer $t$, let $c_t$ be the asymptotic density of natural numbers $n$ for which $s(n + t) \geq s(n)$, where $s(n)$ denotes the sum of digits of $n$ in base $2$. We prove that $c_t > 1/2$ for $t$ in a set of asymptotic density $1$, thus giving a partial solution to a conjecture of T. W. Cusick stating that $c_t > 1/2$ for all t. Interestingly, this problem has several equivalent formulations, for example that the polynomial $X(X + 1)\cdots(X + t - 1)$ has less than $2^t$ zeros modulo $2^{t+1}$. The proof of the main result is based on Chebyshev's inequality and the asymptotic analysis of a trivariate rational function, using methods from analytic combinatorics., Comment: 27 pages

In his book Topics in Analytic Number Theory, Rademacher considered the generating function of partitions into at most $N$ parts, and conjectured certain limits for the coefficients of its partial fraction decomposition. We carry out an asymptotic analysis that disproves this conjecture, thus confirming recent observations of Sills and Zeilberger (Journal of Difference Equations and Applications 19, 2013), who gave strong numerical evidence against the conjecture.