Uniformly de Bruijn sequences and symbolic Diophantine approximation on fractals

TL;DR

This paper explores the connection between infinite de Bruijn sequences and Diophantine approximation on fractals, establishing positive Hausdorff dimension results and implications for intrinsic Dirichlet functions.

Contribution

It develops the theory of infinite de Bruijn sequences to address questions in Diophantine approximation on fractals, showing their Hausdorff dimension is positive.

Findings

Set of infinite de Bruijn sequences has positive Hausdorff dimension.

Sequences relate to Diophantine approximation on fractals.

Optimality of intrinsic Dirichlet functions is established.

Abstract

Intrinsic Diophantine approximation on fractals, such as the Cantor ternary set, was undoubtedly motivated by questions asked by K. Mahler (1984). One of the main goals of this paper is to develop and utilize the theory of infinite de Bruijn sequences in order to answer closely related questions. In particular, we prove that the set of infinite de Bruijn sequences in $k\geq 2$ letters, thought of as a set of real numbers via a decimal expansion, has positive Hausdorff dimension. For a given $k$, these sequences bear a strong connection to Diophantine approximation on certain fractals. In particular, the optimality of an intrinsic Dirichlet function on these fractals with respect to the height function defined by symbolic representations of rationals follows from these results.