On the set of points represented by harmonic subseries

2024/05/13 by Vjekoslav Kovač, Kovač, Vjekoslav · 2 citations
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #Digital Image Processing Techniques #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2405.07681

openalex publication_date 2024/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

We help Alice play a certain "convergence game" against Bob and win the prize, which is a constructive solution to a problem by Erdős and Graham, posed in their 1980 book on open questions in combinatorial number theory. Namely, after several reductions using peculiar arithmetic identities, the game outcome shows that the set of points (∑_{n∈ A}(1)/(n), ∑_{n∈ A}(1)/(n+1), ∑_{n∈ A}(1)/(n+2)), obtained as A ranges over infinite sets of positive integers, has a non-empty interior. This generalizes a two-dimensional result by Erdős and Straus.

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