Number Theory seminar

Kevin Hare, Pure Math Department, University of Waterloo

“Base d expansions with digits 0 to q − 1”

Let d and q be positive integers, and consider representing a positive integer n with base d and digits 0, 1, · · · , q − 1. If q < d, then not all positive integers can be represented. If q = d, every positive integer can be represented in exactly one way. If q > d, then there may be multiple ways of representing the integer n. Let fd,q (n) be the number of representations of n with base d and digits 0, 1, · · · , q − 1. For example, if d = 2 and q = 7 we might represent 6 as (110)2 = 1·22 +1·21 +0·20 as well as (102)2 = 1 · 22 + 0 · 21 + 2 · 20. In fact, there are six representations in this case (110)2, (102)2, (30)2, (22)2, (14)2 and (6)2, hence f2,7(6) = 6. In this talk we will discuss the asymptotics of fd,q(n) as n → ∞. This depends in a rather strange way on the Generalized Thue-Morse sequence. While many results are computationally/experimentally true, only partial results are known.