Joint distribution of leftmost digits in positional notation and Schanuels's conjecture

TL;DR

This paper investigates the joint distribution of leftmost digits of numbers in various bases, establishing conditions for their independence and connecting these to deep conjectures in transcendental number theory.

Contribution

It proves that the joint distribution is surjective only when the logarithms of the bases are rationally independent, and relates this to Schanuel's conjecture for higher dimensions.

Findings

Rational independence of log-bases is necessary for surjectivity.

Converse holds for two bases under certain conjectures.

Links digit distribution properties to transcendental number theory.

Abstract

Assume that $n \geq 2$ and $B = (b_1,...,b_n)$ has distince integer entries $\geq 3.$ For $x > 0$ let $d_B(x) := (d_{b_1}(x),...,d_{b_n}(x))$ where $d_{b_i}(x) \in \{1,...,b_i-1\}$ is the leftmost digit in the base-$b_i$ positional notation representation of $x.$ We prove that if $d_B$ is surjective, then $\ln b_i$ and $\ln b_j$ are rationally independent whenever $i \neq j.$ We prove the converse for $n = 2,$ and for $n \geq 3$ if $\{\ln p : p \mbox{ prime} \}$ is algebraically independent, a condition implied by Schanuel's conjecture about transcendental numbers.