TY - JOUR

T1 - On a problem of erdős and turán and some related results

AU - Alon, Noga

AU - Kolountzakis, Mihail N.

PY - 1995/11

Y1 - 1995/11

N2 - We employ the probabilistic method to prove a stronger version of a result of Helm, related to a conjecture of Erdo{combining double acute accent}s and Turán about additive bases of the positive integers. We show that for a class of random sequences of positive integers A, which satisfy ∣A ∩ [1, x]∣ ≫ √x with probability 1, all integers in any interval [ I, N] can be written in at least c1 log N and at most c2 log N ways as a difference of elements of A ∩ [1, N2]. We also prove several results related to another result of Helm. We show that for every sequence of positive integers M, with counting function M(x), there is always another sequence of positive integers A such that M ∩ (A - A) = ∅ and A(x) > x/(M(x) + 1). We also show that this result is essentially best possible, and we show how to construct a sequence A with A(x) > cx/(M(x) + 1) for which every element of M is represented exactly as many times as we wish as a difference of elements of A.

AB - We employ the probabilistic method to prove a stronger version of a result of Helm, related to a conjecture of Erdo{combining double acute accent}s and Turán about additive bases of the positive integers. We show that for a class of random sequences of positive integers A, which satisfy ∣A ∩ [1, x]∣ ≫ √x with probability 1, all integers in any interval [ I, N] can be written in at least c1 log N and at most c2 log N ways as a difference of elements of A ∩ [1, N2]. We also prove several results related to another result of Helm. We show that for every sequence of positive integers M, with counting function M(x), there is always another sequence of positive integers A such that M ∩ (A - A) = ∅ and A(x) > x/(M(x) + 1). We also show that this result is essentially best possible, and we show how to construct a sequence A with A(x) > cx/(M(x) + 1) for which every element of M is represented exactly as many times as we wish as a difference of elements of A.

UR - http://www.scopus.com/inward/record.url?scp=58149369948&partnerID=8YFLogxK

U2 - 10.1006/jnth.1995.1129

DO - 10.1006/jnth.1995.1129

M3 - מאמר

AN - SCOPUS:58149369948

VL - 55

SP - 82

EP - 93

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 1

ER -