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52
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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Cited by 34 (13 self)
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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.
Concentration of points on two and three dimensional modular hyperbolas and applications. Geometric and Functional Analysis
"... Let p be a large prime number, K,L,M, λ be integers with 1 ≤ M ≤ p and gcd(λ, p) = 1. The aim of our paper is to obtain sharp upper bound estimates for the number I2(M;K,L) of solutions of the congruence xy ≡ λ (mod p), K + 1 ≤ x ≤ K +M, L+ 1 ≤ y ≤ L+M and for the number I3(M;L) of solutions of the ..."
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Cited by 14 (2 self)
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Let p be a large prime number, K,L,M, λ be integers with 1 ≤ M ≤ p and gcd(λ, p) = 1. The aim of our paper is to obtain sharp upper bound estimates for the number I2(M;K,L) of solutions of the congruence xy ≡ λ (mod p), K + 1 ≤ x ≤ K +M, L+ 1 ≤ y ≤ L+M and for the number I3(M;L) of solutions of the congruence
Additive representation in thin sequences, I: Waring’s problem for cubes
 ANN. SCI. ÉCOLE NORM. SUP
, 2001
"... In this paper we investigate representation of numbers from certain thin sequences like the squares or cubes by sums of cubes. It is shown, in particular, that almost all values of an integral cubic polynomial are sums of six cubes. The methods are very flexible and may be applied to many related pr ..."
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Cited by 12 (11 self)
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In this paper we investigate representation of numbers from certain thin sequences like the squares or cubes by sums of cubes. It is shown, in particular, that almost all values of an integral cubic polynomial are sums of six cubes. The methods are very flexible and may be applied to many related problems.
On Diophantine inequalities: Freeman’s asymptotic formulae
 PROCEEDINGS OF THE SESSION IN ANALYTIC NUMBER THEORY AND DIOPHANTINE EQUATIONS
, 2003
"... ..."
The density of rational lines on cubic hypersurfaces
 Trans. Amer. Math. Soc
"... Abstract. We provide a lower bound for the density of rational lines on the hypersurface dened by an additive cubic equation in at least 57 variables. In the process, we obtain a result on the paucity of nontrivial solutions to an associated system of Diophantine equations. 1. ..."
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Cited by 10 (6 self)
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Abstract. We provide a lower bound for the density of rational lines on the hypersurface dened by an additive cubic equation in at least 57 variables. In the process, we obtain a result on the paucity of nontrivial solutions to an associated system of Diophantine equations. 1.
Exceptional sets for Diophantine inequalities
"... Abstract. We apply Freeman’s variant of the DavenportHeilbronn method to investigate the exceptional set of real numbers not close to some value of a given real diagonal form at an integral argument. Under appropriate conditions, we show that the exceptional set in the interval [−N,N] has measure O ..."
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Cited by 5 (1 self)
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Abstract. We apply Freeman’s variant of the DavenportHeilbronn method to investigate the exceptional set of real numbers not close to some value of a given real diagonal form at an integral argument. Under appropriate conditions, we show that the exceptional set in the interval [−N,N] has measure O(N1−δ), for a positive number δ. 1.
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 5 (4 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