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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.
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
"... ABSTRACT. – 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 man ..."
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ABSTRACT. – 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. 2001 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. – Dans cet article nous étudions la représentation des nombres de certaines suites rares comme celles des carrés ou des cubes. Il est démontré notamment que presque toutes les valeurs d’un polynôme de degré trois sont des sommes de six cubes. Ces méthodes, très flexibles, sont applicables à beaucoup de problèmes analogues. 2001 Éditions scientifiques et médicales Elsevier SAS 1.
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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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.
On Diophantine inequalities: Freeman’s asymptotic formulae, Proceedings of the session in analytic number theory and Diophantine equations
 Mathematische Schriften
, 2003
"... 1. Introduction. It is only within the past couple of years that the DavenportHeilbronn method, now in its second halfcentury of life, has delivered asymptotic formulae for the number of solutions of diophantine inequalities in many variables. Let k and s be positive integers with k> 2 and s> ..."
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1. Introduction. It is only within the past couple of years that the DavenportHeilbronn method, now in its second halfcentury of life, has delivered asymptotic formulae for the number of solutions of diophantine inequalities in many variables. Let k and s be positive integers with k> 2 and s> 2k + 1, and let τ be any positive number. Then whenever λ1,..., λs are nonzero real numbers, not all in
WARING’S PROBLEM IN FUNCTION FIELDS
"... Abstract. Let 픽 푞 [푡] denote the ring of polynomials over the finite field 픽 푞 of characteristic 푝, and write 핁푘 푞 [푡] for the additive closure of the set of 푘th powers of polynomials in 픽 푞 [푡]. Define 퐺푞(푘) to be the least integer 푠 satisfying the property that every polynomial in 핁푘 푞 [푡] of suff ..."
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Abstract. Let 픽 푞 [푡] denote the ring of polynomials over the finite field 픽 푞 of characteristic 푝, and write 핁푘 푞 [푡] for the additive closure of the set of 푘th powers of polynomials in 픽 푞 [푡]. Define 퐺푞(푘) to be the least integer 푠 satisfying the property that every polynomial in 핁푘 푞 [푡] of sufficiently large degree admits a strict representation as a sum of 푠 푘th powers. We employ a version of the HardyLittlewood method involving the use of smooth polynomials in order to establish a bound of the shape 퐺푞(푘) ≤ 퐶 푘 log 푘 + 푂( 푘 log log 푘). Here, the coefficient 퐶 is equal to 1 when 푘 < 푝, and 퐶 is given explicitly in terms of 푘 and 푝 when 푘> 푝, but in any case satisfies 퐶 ≤ 4/3. There are associated conclusions for the solubility of diagonal equations over 픽 푞 [푡], and for exceptional set estimates in Waring’s problem. 1. Introduction. A
ON WEYL’S INEQUALITY, HUA’S LEMMA, AND EXPONENTIAL SUMS OVER BINARY FORMS
 VOL. 100, NO. 3 DUKE MATHEMATICAL JOURNAL
, 1999
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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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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.