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Small gaps in coefficients of Lfunctions and Bfree numbers in short intervals
, 2005
"... We discuss questions related to the nonexistence of gaps in the series defining modular forms and other arithmetic functions of various types, and improve results of Serre, Balog & Ono and Alkan using new results about exponential sums and the distribution of Bfree numbers. ..."
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Cited by 4 (3 self)
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We discuss questions related to the nonexistence of gaps in the series defining modular forms and other arithmetic functions of various types, and improve results of Serre, Balog & Ono and Alkan using new results about exponential sums and the distribution of Bfree numbers.
Arbitrarily Tight Bounds On The Distribution Of Smooth Integers
 Proceedings of the Millennial Conference on Number Theory
, 2002
"... This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve ..."
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Cited by 3 (1 self)
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This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.
ON VU’S THIN BASIS THEOREM IN WARING’S PROBLEM
"... V. Vu has recently shown that when k ≥ 2 and s is sufficiently large in terms of k, then there exists a set X(k), whose number of elements up to t is smaller than a constant times (t log t) 1/s, for which all large integers n are represented as the sum of s kth powers of elements of X(k) in order lo ..."
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V. Vu has recently shown that when k ≥ 2 and s is sufficiently large in terms of k, then there exists a set X(k), whose number of elements up to t is smaller than a constant times (t log t) 1/s, for which all large integers n are represented as the sum of s kth powers of elements of X(k) in order log n ways. We establish this conclusion with s ∼ k log k, improving on the constraint implicit in Vu’s work which forces s to be as large as k 4 8 k. Indeed, the methods of this paper show, roughly speaking, that whenever existing methods permit one to show that all large integers are the sum of H(k) kth powers of natural numbers, then H(k) + 2 variables suffice to obtain a corresponding conclusion for “thin sets, ” in the sense of Vu. 1.
WHEN THE SIEVE WORKS
, 2012
"... Abstract. We are interested in classifying those sets of primes P such that when we sieve out the integers up to x by the primes in P c we are left with roughly the expected number of unsieved integers. In particular, we obtain the first general results for sieving an interval of length x with prime ..."
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Abstract. We are interested in classifying those sets of primes P such that when we sieve out the integers up to x by the primes in P c we are left with roughly the expected number of unsieved integers. In particular, we obtain the first general results for sieving an interval of length x with primes including some in ( √ x, x], using methods motivated by additive combinatorics. 1. Introduction and