Small values of signed harmonic sums

TL;DR

This paper investigates how closely signed harmonic sums can approximate any real number, proving that for large N, the minimal distance decreases faster than any exponential of a squared logarithm, revealing deep properties of these sums.

Contribution

It establishes a new bound on the minimal distance of signed harmonic sums from any real number, showing an exponential decay involving the square of the logarithm of N.

Findings

Minimal distance decreases faster than exp(-C*(log N)^2) for large N

The bound holds for any real number τ and positive constant C<1/log 4

Provides insight into the distribution of signed harmonic sums

Abstract

For every $\tau\in\mathbb{R}$ and every integer $N$, let $\mathfrak{m}_N(\tau)$ be the minimum of the distance of $\tau$ from the sums $\sum_{n=1}^N s_n/n$, where $s_1, \ldots, s_n \in \{-1, +1\}$. We prove that $\mathfrak{m}_N(\tau) < \exp\!\big(-C(\log N)^2\big)$, for all sufficiently large positive integers $N$ (depending on $C$ and $\tau$), where $C$ is any positive constant less than $1/\log 4$.