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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. 1

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Erdös and Szemerédi proved that if A is a set of k positive integers, then there must be at least ck1+δ integers that can be written as the sum or product of two elements of A, wherecisaconstant and δ>0. Nathanson proved that the result holds for δ = 1 31 result holds for δ = 1 1 and c = 5 20.. In this paper it is proved that the 1.

Abstract. We give a simple and natural construction of hypergraph regularization. It yields a short proof of a hypergraph regularity lemma. Consequently, as an example of its applications, we have a short self-contained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well as its multidimensional extension by Furstenberg-Katznelson (1978). 1.

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Abstract. Define the Liouville function for A, a subset of the primes P, by λA(n) = (−1) ΩA(n) where ΩA(n) is the number of prime factors of n coming from A counting multiplicity. For the traditional Liouville function, A is the set of all primes. Denote LA(x): = X n≤x λA(n) and RA: = lim n→∞ LA(n) n It is known that for each α ∈ [0, 1] there is an A ⊂ P such that RA = α. Given certain restrictions on the sifting density of A, asymptotic estimates for P n≤x λA(n) can be given. With further restrictions, more can be said. For an odd prime p, define the character–like function λp as λp(pk + i) = (i/p) for i = 1,..., p − 1 and k ≥ 0, and λp(p) = 1 where (i/p) is the Legendre symbol (for example, λ3 is defined by λ3(3k + 1) = 1, λ3(3k + 2) = −1 (k ≥ 0) and λ3(3) = 1) For the partial sums of character–like functions we give exact values and asymptotics; in particular we prove the following theorem. Theorem. If p is an odd prime, then X max n≤x λp(k) ≃ log x. ˛k≤n This result is related to a question of Erdős concerning the existence of bounds for number–theoretic functions. Within the course of discussion, the ratio φ(n)/σ(n) is considered. 1.

Abstract. A finite set A of integers is square-sum-free if there is no subset of A sums up to a square. In 1986, Erdős posed the problem of determining the largest cardinality of a square-sum-free subset of {1,..., n}. Answering this question, we show that this maximum cardinality is of order n 1/3+o(1). 1.