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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.
The number of cycles of specified normalized length in permutations
 In preparation
"... Abstract. We compute the limiting distribution, as n → ∞, of the number of cycles of length between γn and δn in a permutation of [n] chosen uniformly at random, for constants γ, δ such that 1/(k + 1) ≤ γ < δ ≤ 1/k for some integer k. This distribution is supported on {0, 1,..., k} and has 0th, ..."
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Abstract. We compute the limiting distribution, as n → ∞, of the number of cycles of length between γn and δn in a permutation of [n] chosen uniformly at random, for constants γ, δ such that 1/(k + 1) ≤ γ < δ ≤ 1/k for some integer k. This distribution is supported on {0, 1,..., k} and has 0th, 1st,..., kth moments equal. For more general choices of γ, δ we show that such a limiting distribution exists, which can be given explicitly in terms of certain integrals over intersections of hypercubes with halfspaces; these integrals are analytically intractable but a recurrence specifying them can to those of a Poisson distribution with parameter log δ γ be given. The results herein provide a basis of comparison for similar statistics on restricted classes of permutations. The distribution of the number of kcycles in a permutation of [n], for a fixed k, converges to a Poisson distribution with mean 1/k as k → ∞. In particular the mean number of kcycles and the variance of the number of kcycles are both 1/k whenever n ≥ k and n ≥ 2k respectively. If instead of holding k constant we let it vary with n, the number of αncycles in permutations of [n] approaches zero as n → ∞ with α fixed. So to investigate the number of cycles of long lengths, we must rescale and look at many cycle lengths at once. In particular, we consider the number of cycles with length in some interval [γn, δn] as n → ∞. The expectation of the number of cycles with length in this 1/k, which approaches the constant log δ/γ as n grows large. By analogy with the fixedk case we might expect the number of cycles with length in this interval to be Poissondistributed. But this cannot be the case, because there is room for at most 1/γ cycles of length at least γn, and the Poisson distribution can take arbitrarily large values. In the case where 1/γ and 1/δ lie in the same interval [1/(k + 1), 1/k] for some integer k, the limit distribution has the same first k moments as Poisson(log δ/γ). For general γ and δ the situation is considerably more complex but a limit distribution still exists. interval is ∑ δn k=γn
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