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ISBN-10: 089871558X

ISBN-13: 9780898715583

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Extra resources for Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms

Example text

1) shows both that sup α∈m(X 1/3 ) |h(α)| X 5/6 (log X)4 , October 6, 2009 13:49 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 ADDITIVE REPRESENTATION IN THIN SEQUENCES 45 and, whenever Q ≤ X 1/3 , that |h(α)| sup XQ−1/2 (log X)4 . 18) λ1 |p1 −p2 |<τ p1 ,p2 ≤X the first of these estimates yields |h(λ1 α)h(λ2 α)|2 dτ α X 8/3 (log X)9 , m2 and the same is true for the contribution from m1 , by symmetry. 3. 3, we find that c∩K(X 1/3 ,X 1/3 ) |h(λ1 α)h(λ2 α)|2 dτ α X 8/3+ε , an estimate that may also be found on p.

10. The notation used in this memoir is standard, or otherwise explained at the appropriate stage of the proceedings. We write e(α) = exp(2πiα). The distance of a real number α to the nearest integer is α . The integer part of α is [α], and α is the smallest integer n with n ≥ α. We apply the following convention concerning the letter ε. Whenever ε occurs in a statement, it is asserted that this statement is true for all real ε > 0, but constants implicit in Landau or Vinogradov symbols may depend on the actual value of ε.

WOOLEY For comparison, Parsell [30] works under the weaker hypothesis that λ1 /λ2 is irrational, and obtains a result that is essentially equivalent to N σ(τ, ν) − 0 2τ ν 2 dν = o(N 3 ). 7), as well as an improvement when λ1 /λ2 is algebraic, but not by a power of N . Limitations arise from our current knowledge concerning the zeros of the Riemann zeta function. 7) is O(N 3 exp(−c log N )) for some c > 0, and with only moderate extra effort one obtains a saving that corresponds to the sharpest one currently known in the error term for the prime number theorem.

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Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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