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Application of a method of Szemeredi

Published online by Cambridge University Press:  18 May 2009

H. Halberstam
H. Halberstam
Affiliation:
University of Illinois at Urbana-Champaign, Department of Mathematics, 1409 West Green Street, Urbana, Illinois 61801, U.S.A.
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Let ℬ = {bi:b1 <b2<…} be an infinite sequence of positive integers that exceed 1 and are pairwise coprime, so that

Assume also that

Let A=Aℬ denote the sequence of ℬ-free numbers, that is, of positive integers divisible by no element of ℬ. This concept, generalizing square-free and k-free numbers, derives from Erdös [2] who proved in 1966 that there exists a constant c, 0<c<l, independent of ℬ, such that the interval (x, x+xc) contains elements of A provided only that x is large enough. This result of Erdös was shown by Szemeredi [7] in 1973 to hold with c=½+ε, if x≥xo(ε, ℬ), and quite recently Bantle and Grupp [1] have sharpened Szemeredi's result to c=9/20+ε.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 27 , October 1985 , pp. 81 - 85
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

REFERENCES

1.Bantle, G. and Grupp, F., On a problem of Erdos and Szemeredi, /. Number Theory, to appear.Google Scholar
2.Erdös, P., On the difference of consecutive terms of sequences defined by divisibility properties, Ada Arith. 12 (1966), 175–182.CrossRefGoogle Scholar
3.Heath-Brown, D. R., The least square-free number in an arithmetic progression, J. Reine Angew. Math. 332 (1982), 204–220.Google Scholar
4.Narlikar, H. J. and Ramachandra, K., Contributions to the Erdös-Szemeredi theory of sieved integers, Ada Arith. 38 (1980), 157–165.CrossRefGoogle Scholar
5.Szemeredi, E., On the difference of consecutive terms of sequences defined by divisibility properties II, Acta Arith. 23 (1973), 359–361.CrossRefGoogle Scholar