Distribution modulo 1 and the lexicographic world

TL;DR

This paper characterizes the minimal intervals that contain all fractional parts of the form ^n for a positive real number , providing a comprehensive understanding of their distribution.

Contribution

It offers a complete description of the minimal intervals containing all fractional parts ^n for any positive real , advancing the understanding of their distribution modulo 1.

Findings

Complete description of minimal intervals for fractional parts

Characterization of distribution of ^n modulo 1

Insights into the structure of fractional parts for powers of 2

Abstract

We give a complete description of the minimal intervals containing all fractional parts $\{\xi 2^n\}$, for some positive real number $\xi$, and for all $n \geq 0$.