[Paper Review] On the Erd\H{o}s Covering Problem: the density of the uncovered set

Paul Balister, Béla Bollobás+3 more|arXiv (Cornell University)|Nov 8, 2018

Limits and Structures in Graph Theory5 references4 citations

TL;DR

This paper resolves long-standing conjectures in covering systems by introducing a novel method to bound the density of the uncovered set in integer coverings. It establishes that if moduli are sufficiently large and satisfy a weighted reciprocal sum condition involving a specific multiplicative function μ, then the uncovered density is bounded below by $ e^{-4C}/2 $, proving Schinzel's 1967 conjecture and refining Erdős–Graham-type bounds with optimal conditions.

ABSTRACT

Since their introduction by Erd\\H{o}s in 1950, covering systems (that is, finite collections of arithmetic progressions that cover the integers) have been extensively studied, and numerous questions and conjectures have been posed regarding the existence of covering systems with various properties. In particular, Erd\\H{o}s asked if the moduli can be distinct and all arbitrarily large, Erd\\H{o}s and Selfridge asked if the moduli can be distinct and all odd, and Schinzel conjectured that in any covering system there exists a pair of moduli, one of which divides the other. Another beautiful conjecture, proposed by Erd\\H{o}s and Graham in 1980, states that if the moduli are distinct elements of the interval $[n,Cn]$, and $n$ is sufficiently large, then the density of integers uncovered by the union is bounded below by a constant (depending only on $C$). This conjecture was confirmed (in a strong form) by Filaseta, Ford, Konyagin, Pomerance and Yu in 2007, who moreover asked whether the same conclusion holds if the moduli are distinct and sufficiently large, and $\\sum_{i=1}^k \\frac{1}{d_i} < C$. Although this condition turns out not to be sufficiently strong to imply the desired conclusion, as the main result of this paper we will give an essentially best possible condition which is sufficient. Our method has a number of further applications. Most importantly, we prove the conjecture of Schinzel stated above, which was made in 1967. We moreover give an alternative (somewhat simpler) proof of a breakthrough result of Hough, who resolved Erd\\H{o}s' minimum modulus problem, with an improved bound on the smallest difference. Finally, we make further progress on the problem of Erd\\H{o}s and Selfridge.

Motivation & Objective

• To resolve Schinzel's 1967 conjecture that in any covering system with distinct moduli, one modulus divides another.
• To determine the optimal condition on moduli that guarantees a positive lower bound on the density of the uncovered set in covering systems.
• To improve upon the Erdős–Graham conjecture by identifying a condition on the weighted sum $ \sum \mu(d)/d $ that ensures a uniform lower bound on uncovered density.
• To provide a simplified and powerful method for analyzing the density of uncovered sets in covering systems, applicable to multiple longstanding problems.

Proposed method

• Constructs a recursive tree-like structure of arithmetic progressions over nested residue rings, using carefully chosen large primes at each level.
• Defines a multiplicative function $ \mu(p^i) = 1 + (\log p)^{3+\varepsilon}/p $ to weight the moduli, ensuring the sum $ \sum \mu(d)/d \leq C $ while controlling the uncovered density.
• Uses probabilistic bounds via product estimates over primes to show that the uncovered set density decays as $ e^{(2-t)n} $, which tends to zero as depth $ n \to \infty $, under controlled conditions.
• Applies inclusion-exclusion and Euler product estimates to bound the density of uncovered residues at each level of the construction.
• Establishes that the total sum $ \sum \mu(d)/d \leq 2t^2 $, with $ t \in (1, e) $, ensuring the condition is satisfied for arbitrary $ C $, and derives the final bound $ e^{-4C}/2 $.
• Adapts and simplifies Hough’s method for the minimum modulus problem, enabling stronger conclusions about both density and divisibility structure.

Experimental results

Research questions

• RQ1 Can a lower bound on the density of the uncovered set in a covering system be guaranteed solely by the sum $ \sum \mu(d)/d \leq C $, where $ \mu $ is a specific multiplicative function?
• RQ2 Does every covering system with distinct moduli contain a pair where one modulus divides the other? (Schinzel's conjecture)
• RQ3 Is the Erdős–Graham condition on moduli in $[n, Cn]$ necessary and sufficient for a positive lower bound on uncovered density?
• RQ4 Can the method used to resolve the minimum modulus problem be adapted to yield stronger bounds on the density of uncovered sets?
• RQ5 What is the optimal condition on the moduli of a covering system that ensures a uniform lower bound on the density of the uncovered set?

Key findings

• The paper proves Schinzel's conjecture: in any covering system with distinct moduli, one modulus divides another.
• It establishes a sharp lower bound of $ e^{-4C}/2 $ on the density of the uncovered set, provided all moduli are sufficiently large and $ \sum \mu(d)/d \leq C $, where $ \mu(p^i) = 1 + (\log p)^{3+\varepsilon}/p $.
• The authors construct a sequence of covering systems with arbitrarily large moduli and $ \sum 1/d_i < 1 $, yet the uncovered density tends to zero, showing that the reciprocal sum condition alone is insufficient.
• They show that no such lower bound depending only on $ C $ exists if $ \mu(p^i) $ is replaced by $ 1 + O(1/p) $, demonstrating the sharpness of their function choice.
• The method provides a simpler proof of Hough’s breakthrough result on the minimum modulus problem, with an improved bound on the smallest modulus in any covering system with distinct moduli.
• The approach yields new progress on the Erdős–Selfridge problem, showing that the structure of moduli in covering systems is highly constrained when moduli are large and their weighted reciprocal sum is bounded.