Integer Sets of Large Harmonic Sum Which Avoid Long Arithmetic Progressions

TL;DR

This paper investigates digit-restricted integer sets that avoid long arithmetic progressions and demonstrates how their harmonic sums can surpass previous constructions through efficient computation and large-scale search.

Contribution

It introduces conditions for constructing digit-restricted sets avoiding long arithmetic progressions and shows these sets can have larger harmonic sums than earlier methods.

Findings

Identified integer sets avoiding 4- and 10-term arithmetic progressions.

Harmonic sums of these sets exceed previous greedy constructions.

Efficient computation methods for these sets and their sums.

Abstract

We give conditions under which certain digit-restricted integer sets avoid $k$-term arithmetic progressions. These sets and their harmonic sums can be computed efficiently. Through large-scale search, we identify integer sets avoiding arithmetic progressions of length 4 and 10 whose harmonic sums exceed earlier "greedy" constructions.