An amusing sequence of functions

TL;DR

This paper studies a sequence of functions defined by a sum involving sine functions and reveals that all rational points with denominators up to the square root of n become strict local minima as n increases.

Contribution

It demonstrates a novel property of the sequence where all rationals with bounded denominators are strict local minima, linking number theory and analysis.

Findings

All rationals with denominator ≤ √n are strict local minima of f_n.

The sequence exhibits a structured pattern of minima at rational points.

The property holds for all sufficiently large n.

Abstract

We consider the amusing sequence of functions $f_n: \mathbb{R} \rightarrow \mathbb{R}$ given by $$ f_n(x) = \sum_{k=1}^{n}{\frac{|\sin{(k \pi x)}|}{k}}.$$ Every rational point is eventually the location of a strict local minimum of $f_n$: more precisely, $f_n$ has a strict local minimum in all rational points $x=p/q \in \mathbb{Q}$ with $|q| \leq \sqrt{n}$.