Digitally defined sequences and substitutions

Abstract

This thesis is concerned with the joint behaviour of the sum of digits function in two different integer bases as well as uniformity properties of sum of digits functions of generalised numeration systems and substitutions. Gowers norm estimates and their applications play an important role throughout this thesis. First, we prove new Gowers norm estimates for the sum of digits function of special linearly recurrent numeration systems and for the sum of digits function of Ostrowski numeration systems. This extends earlier work by Toumi, who showed that the sum of digits function in base q is Gowers uniform and by Drmota, Müllner and Spiegelhofer, who showed that the sum of digits function of the Zeckendorf numeration system is Gowers uniform. Next, we study s_q(dn+a), the q-ary sum of digits function along sparse arithmetic progressions. Toumi, extending earlier work by Spiegelhofer, showed that the sum of digits function is on average equidistributed modulo m along very sparse arithmetic progressions, i.e., d\sim N^H, for some large constant H. We show that we can remove the averaging over the arithmetic progressions if we restrict to a certain generic class. We even show that the joint distribution of the sum of digits function for multiplicatively independent bases is equidistributed modulo m along these arithmetic progressions. Additionally, we show that the values of s_q(dn+a) are concentrated around the expected value. The third chapter is concerned with the relationship between the values of the sum of digits function in two different multiplicatively independent bases. We study the folklore problem of how many solutions the equation as_{q_1}(n)=bs_{q_2}(n) has for any positive integers a,b and for any integer bases q_1, q_2. Lower bounds for the number of solutions have only been shown in the case a=b=1, q_1=2 and q_2=3 by Spiegelhofer. We provide a full characterisation of when the equation has infinitely many and when it has only finitely many solutions and give a partial result also in the case of three and more bases. We use the results from the previous chapter in key steps in the proof. Lastly, we use the base p representation of natural numbers to study divisibility properties of n! and of binomial coefficients. We show that the lowest nonzero digit of n! in base 12 takes any value of 1 to 11 infinitely many times, thus proving that 12 satisfies the full-range property. On the topic of binomial coefficients, we recall that Singmaster showed that for every integer d, almost all binomial coefficients are divisible by d. The size and structure of the exceptional set are well understood in the case where d is a prime power. We provide the first non-trivial upper bound on the number of binomial coefficients coprime to 6.