Rotation Remainders

TL;DR

This paper investigates a number array called 'the triangle' generated by cyclic rotations and appends, revealing that positions are determined by rotation remainders which form a unique base m/(m+1) expansion in an m-adic ring, leading to properties like aperiodicity.

Contribution

It introduces a novel array construction based on cyclic rotations and characterizes positions using a new type of expansion in an m-adic ring, connecting combinatorial structure with number theory.

Findings

Rotation remainders uniquely determine positions in the triangle.

Rotation remainders form a base m/(m+1) expansion in an m-adic ring.

Sequences of rotation remainders are proven to be aperiodic.

Abstract

We study properties of an array of numbers, called "the triangle," in which each row is formed by rotating all the numbers in the previous row to the left by $m$ positions in cyclical fashion, then appending a number to the end of the row. We show that a number's position in the triangle is uniquely determined by the infinite sequence of column positions--called "rotation remainders"--which we track as the number repeatedly rotates back to the first $m$ columns. The rotation remainders can be viewed as the digits in a "base $m/(m+1)$" expansion in an "$m$-adic" topological ring of a number encoding a given position in the triangle. Properties of these expansions are used to prove interesting claims about the triangle, such as the aperiodicity of any sequence of rotation remainders.