Boolean Function Analogs of Covering Systems

TL;DR

This paper introduces Boolean function analogs of covering systems and explores the problem of covering hyper-boxes with lower-dimensional subboxes, highlighting that primes may often be misleading in such problems.

Contribution

It extends the concept of covering systems to Boolean functions and hyper-boxes, providing new perspectives inspired by Hough's disproof of Erdős's conjecture.

Findings

Boolean function analogs of covering systems are developed.

Primes are often irrelevant in covering problems.

The approach may have implications for understanding the Riemann Hypothesis.

Abstract

Bob Hough recently disproved a long-standing conjecture of Paul Erd\H{o}s regarding covering systems. Inspired by his seminal paper, we describe analogs of covering systems to Boolean functions, and more generally, the problem of covering discrete hyper-boxes by non-parallel lower dimensional hyper-subboxes. We point out that very often primes are red herrings. This is definitely the case for covering system, and who knows, perhaps also for the Riemann Hypothesis.