Patterns in rational base number systems

TL;DR

This paper investigates digit patterns in rational base number systems, demonstrating their uniform distribution, analyzing sum-of-digits functions, and constructing normal numbers using advanced tools like self-affine tiles and Fourier analysis.

Contribution

It proves uniform digit distribution and normality in rational base systems using novel methods involving self-affine tiles and adèle Fourier analysis.

Findings

Digits are uniformly distributed in rational base representations.

Constructs normal numbers in base a using rational base representations.

Uses advanced mathematical tools to analyze non-context-free languages.

Abstract

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.