# Dedekind sums take each value infinitely many times

## TL;DR

This paper proves that each value of the Dedekind sum is attained infinitely often by constructing sequences of pairs with increasing denominator, demonstrating the sum's value distribution is dense in a certain sense.

## Contribution

It establishes that every Dedekind sum value occurs infinitely many times with unbounded denominators, revealing new insights into the sum's value distribution.

## Key findings

- Dedekind sums take each value infinitely many times
- Sequences of pairs with increasing denominators can produce the same sum value
- The distribution of Dedekind sum values is dense in a certain sense

## Abstract

For $a\in \Bbb Z$ and $b\in\Bbb N$, $(a,b)=1$, let $s(a,b)$ denote the classical Dedekind sum. We show that Dedekind sums take this value infinitely many times in the following sense. There are pairs $(a_i,b_i)$, $i\in\Bbb N$, with $b_i$ tending to infinity as $i$ grows, such that $s(a_i,b_i)=s(a,b)$ for all $i\in \Bbb N$.