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18
SIEVING BY LARGE INTEGERS AND COVERING SYSTEMS OF CONGRUENCES
, 2006
"... 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 confirm ..."
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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
Necessary conditions for distinct covering systems with squarefree moduli
 ACTA ARITHMETICA LIX.L (1991)
, 1991
"... ..."
On sums and products of integers
 PROC. AMER. MATH. SOC
, 1999
"... 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.. ..."
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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.
A simple regularization of hypergraphs
"... 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 selfcontained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well a ..."
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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 selfcontained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well as its multidimensional extension by FurstenbergKatznelson (1978). 1.
Sets of integers and quasiintegers with pairwise common divisor
 Acta Arith
, 1996
"... u ∈ N: u, s−1 ..."
On extremal sets without coprimes
"... We use the following notations: Z denotes the set of all integers, N denotes the set of positive integers, and P = {p1,p2,...} = {2,3,5,...} denotes the set of all primes. We set ..."
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We use the following notations: Z denotes the set of all integers, N denotes the set of positive integers, and P = {p1,p2,...} = {2,3,5,...} denotes the set of all primes. We set
COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN {−1, 1}
"... 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) ..."
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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.