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.  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 ✩

The Hasse principle for pairs of diagonal cubic forms

“…Next, in section 3, we introduce a new method for averaging Fourier coefficients over thin sequences, and we apply it to establish Theorem 3. Though motivated by recent work of Wooley [25] and Brüdern, Kawada and Wooley [6], this section contains the most novel material in this paper. In section 4, we derive Theorem 4 as well as some other mean value estimates that all follow from Theorem 3.…”

Additive representation in thin sequences, V: Mixed problems of Waring's type

“…In the first part of this series of papers (see Brüdern, Kawada and Wooley [2]), we introduced an approach to additive problems in which one seeks to establish that almost all natural numbers in some fixed polynomial sequence are represented in a prescribed manner, thereby deriving non-trivial estimates for exceptional sets in thin sequences. We illustrated our methods by obtaining upper bounds for the exceptional sets associated with the representation of integers from quadratic, or cubic, polynomial sequences by sums of six cubes of positive integers.…”

Subconvexity for additive equations: Pairs of undenary cubic forms

“…By means of a change of variable, one discerns from (3.3) and Lemma 2.2 that 6) and in an analogous manner one obtains J H i,l B 3 . Finally, we put…”