Sieving by Large Integers and Covering Systems of Congruences

An old question of Erdős asks if there exists, for each number N , a finite set S of integers greater than N and residue classes r(n) (mod n) for n ∈ S whose union is Z. We prove that if ∑ n∈S 1/n is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erdős and Selfridge. We also prove a conjecture of Erdős and Graham, that, for each fixed number K > 1, the complement in Z of any union of residue classes r(n) (mod n), for distinct n ∈ (N,KN ], has density at least dK for N sufficiently large. Here dK is a positive number depending only on K . Either of these new results implies another conjecture of Erdős and Graham, that if S is a finite set of moduli greater than N , with a choice for residue classes r(n) (mod n) for n ∈ S which covers Z, then the largest member of S cannot be O(N). We further obtain stronger forms of these results and establish other information, including an improvement of a related theorem of Haight.