Geodesic
Application of a method of Szemeredi
Glasgow mathematical journal, Tome 26 (1985), pp. 81-85

Voir la notice de l'article provenant de la source Cambridge University Press

Let B = {bi:b1 <b2<...} be an infinite sequence of positive integers that exceed 1 and are pairwise coprime, so thatAssume also thatLet 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=1⁄2+ε, if x≥xo(ε, B), and quite recently Bantle and Grupp [1] have sharpened Szemeredi's result to c=9/20+ε. 
Halberstam, H. Application of a method of Szemeredi. Glasgow mathematical journal, Tome 26 (1985), pp. 81-85. doi: 10.1017/S001708950000608X
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[1] 1.Bantle, G. and Grupp, F., On a problem of Erdos and Szemeredi, /. Number Theory, to appear. Google Scholar

[2] 2.Erdös, P., On the difference of consecutive terms of sequences defined by divisibility properties, Ada Arith. 12 (1966), 175–182. Google Scholar | DOI

[3] 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] 4.Narlikar, H. J. and Ramachandra, K., Contributions to the Erdös-Szemeredi theory of sieved integers, Ada Arith. 38 (1980), 157–165. Google Scholar | DOI

[5] 5.Szemeredi, E., On the difference of consecutive terms of sequences defined by divisibility properties II, Acta Arith. 23 (1973), 359–361. Google Scholar | DOI

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