Jörg Brüdern

On Waring's problem: three cubes and a sixth power

The natural interpretation of even moments of exponential sums, in terms of the number of solutions of certain underlying diophantine equations, permits a rich interplay to be developed between simple analytic inequalities, and estimates for those even moments. This interplay is in large part responsible for the remarkable success enjoyed by the Hardy-Littlewood method in its application to numerous problems of additive type. In the absence of such an interpretation, the most effective method for bounding odd and fractional moments, hitherto, has been to apply H61ders inequality to interpolate linearly between the exponents arising at even moments. The object of this paper is to establish a method for handling all moments of exponential sums over smooth numbers non-trivially, thereby breaking out of the latter (classically) implied convex region of permissible exponents. In view of the great flexibility and applicability of the new iterative methods of Vaughan and Wooley (see, for example, [13, 16, 18]), this breakthrough has many consequences. In this paper we confine ourselves to two relatively accessible applications, deriving new bounds for sums of cubes, and strengthening substantially what is known about quasi-diagonal behaviour.The natural interpretation of even moments of exponential sums, in terms of the number of solutions of certain underlying diophantine equations, permits a rich interplay to be developed between simple analytic inequalities, and estimates for those even moments. This interplay is in large part responsible for the remarkable success enjoyed by the Hardy-Littlewood method in its application to numerous problems of additive type. In the absence of such an interpretation, the most effective method for bounding odd and fractional moments, hitherto, has been to apply H61ders inequality to interpolate linearly between the exponents arising at even moments. The object of this paper is to establish a method for handling all moments of exponential sums over smooth numbers non-trivially, thereby breaking out of the latter (classically) implied convex region of permissible exponents. In view of the great flexibility and applicability of the new iterative methods of Vaughan and Wooley (see, for example, [13, 16, 18]), this breakthrough has many consequences. In this paper we confine ourselves to two relatively accessible applications, deriving new bounds for sums of cubes, and strengthening substantially what is known about quasi-diagonal behaviour. In order to describe the consequences of our new method for mean values of smooth Weyl sums, we shall require some notation. Denote by ,~(P,R) the set of R-smooth numbers of size at most P, that is

On a certain senary cubic form

A strong form of the Manin-Peyre conjecture with a power-saving error term is proved for a certain cubic fourfold.

On the exponential sum over k-free numbers

This paper is concerned with mean values of exponential sum generating functions over k–free numbers, and especially their L1–means. We also provide non–trivial estimates for the L1–means of such generating functions restricted to the minor arcs occurring in Hardy–Littlewood dissections, thereby permitting the circle method to be successfully applied to certain additive problems hitherto beyond its reach.

Additive representation in thin sequences, I: Waring's problem for cubes

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.

ON WARING'S PROBLEM FOR CUBES AND SMOOTH WEYL SUMS

Non-trivial estimates for fractional moments of smooth cubic Weyl sums are developed. Complemented by bounds for such sums of use on both the major and minor arcs in a Hardy--Littlewood dissection, these estimates are applied to derive an upper bound for the

Analytic Number Theory

s

The Hasse principle for pairs of diagonal cubic forms

th moment of the smooth cubic Weyl sum of the expected order of magnitude as soon as

A THREE SQUARES THEOREM WITH ALMOST PRIMES

s\ge 7.691

A Ternary Problem in Additive Prime Number Theory

. Related arguments demonstrate that all large integers

Sums of four cubes

n