Thue-Morse at Multiples of an Integer

Let (t_n) be the classical Thue-Morse sequence defined by t_n = s_2(n) (mod 2), where s_2 is the sum of the bits in the binary representation of n. It is well known that for any integer k>=1 the frequency of the letter "1" in the subsequence t_0, t_k, t_{2k}, ... is asymptotically 1/2. Here we prove that for any k there is a n<=k+4 such that t_{kn}=1. Moreover, we show that n can be chosen to have Hamming weight <=3. This is best in a twofold sense. First, there are infinitely many k such that t_{kn}=1 implies that n has Hamming weight >=3. Second, we characterize all k where the minimal n equals k, k+1, k+2, k+3, or k+4. Finally, we present some results and conjectures for the generalized problem, where s_2 is replaced by s_b for an arbitrary base b>=2.Comment: 14 page

Infinite Systems of Functional Equations and Gaussian Limiting Distributions

In this paper infinite systems of functional equations in finitely or infinitely many random variables arising in combinatorial enumeration problems are studied. We prove sufficient conditions under which the combinatorial random variables encoded in the generating function of the system tend to a finite or infinite dimensional limiting distribution

Patterns in rational base number systems

Number systems with a rational number $a/b > 1$ as base have gained interest in recent years. In particular, relations to Mahler's 3/2-problem as well as the Josephus problem have been established. In the present paper we show that the patterns of digits in the representations of positive integers in such a number system are uniformly distributed. We study the sum-of-digits function of number systems with rational base $a/b$ and use representations w.r.t. this base to construct normal numbers in base $a$ in the spirit of Champernowne. The main challenge in our proofs comes from the fact that the language of the representations of integers in these number systems is not context-free. The intricacy of this language makes it impossible to prove our results along classical lines. In particular, we use self-affine tiles that are defined in certain subrings of the ad\'ele ring $\mathbb{A}_\mathbb{Q}$ and Fourier analysis in $\mathbb{A}_\mathbb{Q}$. With help of these tools we are able to reformulate our results as estimation problems for character sums

Square root singularities of infinite systems of functional equations

Infinite systems of equations appear naturally in combinatorial counting problems. Formally, we consider functional equations of the form y(x) = F(x,y(x)), where F(x,y) : C×ℓ p → ℓ p is a positive and nonlinear function, and analyze the behavior of the solution y(x) at the boundary of the domain of convergence. In contrast to the finite dimensional case different types of singularities are possible. We show that if the Jacobian operator of the function F is compact, then the occurring singularities are of square root type, as it is in the finite dimensional setting. This leads to asymptotic expansions of the Taylor coefficients of y(x)

Square root singularities of infinite systems of functional equations

Infinite systems of equations appear naturally in combinatorial counting problems. Formally, we consider functional equations of the form $\mathbf{y} (x)=F(x,\mathbf{y} (x))$, where $F(x,\mathbf{y} ):\mathbb{C} \times \ell^p \to \ell^p$ is a positive and nonlinear function, and analyze the behavior of the solution $\mathbf{y} (x)$ at the boundary of the domain of convergence. In contrast to the finite dimensional case different types of singularities are possible. We show that if the Jacobian operator of the function $F$ is compact, then the occurring singularities are of square root type, as it is in the finite dimensional setting. This leads to asymptotic expansions of the Taylor coefficients of $\mathbf{y} (x)$

Sur un problème de Chen et Liu concernant la factorisation en puissances des premiers de n!n!

International audienceFor a fixed prime p, let e_p(n!) denote the order of p in the prime factorization of n!. Chen and Liu (2007) asked whether for any fixed m, one has {e_p(n^2 !) mod m : n ∈ Z} = Z_m and {e_p(q!) mod m : q prime} = Z_m. We answer these two questions and show asymptotic formulas for #{n >x^(4/(3h+1))