Rotation Remainders

Abstract

An object in mathematics that first appears for the sake of amusement can demonstrate deep connections with well-known questions about numbers. Consider an array of numbers formed by a rotating queue: starting with just the number 1, to obtain the next row we move everything in the last row m steps to the left, with numbers at the front of the row cycling around and appearing at the back. We then append 1 plus the head of the last row to the new row. Here is an example showing the first few rows when m = 3. . . . It is obvious that each number will repeatedly rotate back to one of the first m columns. Suppose we label the columns 0, 1, 2, . .. and we want to track the following behaviors: (1) the column positions in which a number appears as it repeatedly moves back to the first m columns, (2) the frequency with which 1 is at the head of a row, (3) the frequency with which a new number appears in the array, and (4) the frequency with which a certain number appears in a row. This paper will explore these questions, with the largest amount of work being spent on the first question. In this introduction, we will introduce some notation to get us started. Then we will give our main results, which will be proved in the main body of the paper. [From Introduction]