Ernest Croot

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The large sieve has its origins in the work of Linnik and Rényi. It was developed to deal with sequences that avoid a positive proportion of residue classes. It was later simplified by Roth, Bombieri, Davenport, Halberstam, Montgomery, Gallagher and many others. For a survey see Montgomery [12] and Bombieri [1]. It is known that Montgomery’s large sieve(More)
Let L(c, x) = e √ log x log log . We prove that if a1 (mod q1), ..., ak (mod qk) are a maximal collection of non-intersecting arithmetic progressions, with 2 ≤ q1 < q2 < · · · < qk ≤ x, then x L( √ 2 + o(1), x) < k < x L(1/6− o(1), x) . In the case for when the qi’s are square-free, we obtain the improved upper bound k < x L(1/2− o(1), x) .
In this paper we will use the following notation. Given a set of positive integers S, we let S(x) denote the number of elements in S that are ≤ x, and we let |S| denote the total number of elements of S. Given two sets of positive integers A and B, we denote the sumset {a + b : a ∈ A, b ∈ B} by A + B; and so, the number of elements in A + B that are ≤ x(More)
We further examine Kummer's incorrect conjectured asymptotic estimate for the size of the rst factor of the class number of a cyclotomic eld, h 1 (p). Whereas Kummer had conjectured that h 1 (p) G(p) := 2p(p=4 2) p?1 4 we show, under certain plausible assumptions, that there exist constants a ; b such that h 1 (p) G(p) for a x= log b x primes p x whenever(More)