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13
An invitation to additive prime number theory
, 2004
"... The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sam ..."
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The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.
Equal sums of like polynomials
 Bull. London Math. Soc
"... Let f ∈ Z[x] be a polynomial of degree d. We establish the paucity of nontrivial positive integer solutions to the equation f(x1) + f(x2) = f(x3) + f(x4), provided that d ≥ 7. We also investigate the corresponding situation for equal sums of three like polynomials. Mathematics Subject Classificati ..."
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Cited by 4 (3 self)
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Let f ∈ Z[x] be a polynomial of degree d. We establish the paucity of nontrivial positive integer solutions to the equation f(x1) + f(x2) = f(x3) + f(x4), provided that d ≥ 7. We also investigate the corresponding situation for equal sums of three like polynomials. Mathematics Subject Classification (2000): 11D45 (11P05) 1
Vinogradov’s mean value theorem via efficient congruencing
"... Abstract. We obtain estimates for Vinogradov’s integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring’s problem holds for sums of s kth powers of natural ..."
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Abstract. We obtain estimates for Vinogradov’s integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring’s problem holds for sums of s kth powers of natural numbers whenever s � 2k 2 + 2k − 3. 1.
ON WEYL’S INEQUALITY, HUA’S LEMMA, AND EXPONENTIAL SUMS OVER BINARY FORMS
 VOL. 100, NO. 3 DUKE MATHEMATICAL JOURNAL
, 1999
"... ..."
Waring’s Problem: A Survey
"... tribus, &c. usque ad novemdecim compositus, & sic deinceps.” ..."
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tribus, &c. usque ad novemdecim compositus, & sic deinceps.”
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.
Diophantine Methods for Exponential Sums, and Exponential Sums for Diophantine Problems
, 2003
"... Recent developments in the theory and application of the HardyLittlewood method are discussed, concentrating on aspects associated with diagonal diophantine problems. Recent efficient differencing methods for estimating mean values of exponential sums are described first, concentrating on developme ..."
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Recent developments in the theory and application of the HardyLittlewood method are discussed, concentrating on aspects associated with diagonal diophantine problems. Recent efficient differencing methods for estimating mean values of exponential sums are described first, concentrating on developments involving smooth Weyl sums. Next, arithmetic variants of classical inequalities of Bessel and CauchySchwarz are discussed. Finally, some emerging connections between the circle method and arithmetic geometry are mentioned.
MEAN VALUE ESTIMATES AND APPLICATIONS IN ARITHMETIC COMBINATORICS
, 2012
"... in accordance with the requirements of the degree ..."