Trevor D. Wooley

Vinogradov's mean value theorem via efficient congruencing

We obtain estimates for Vinogradov’s integral that for the rst 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.

Vinogradov’s mean value theorem via efficient congruencing, II

We apply the efficient congruencing method to estimate Vinogradovs integral for moments of order 2s, with 1 =k^2-1. In this way we come half way to proving the main conjecture in two different directions. There are consequences for estimates of Weyl type, and in several allied applications. Thus, for example, the anticipated asymptotic formula in Warings problem is established for sums of s kth powers of natural numbers whenever s>=2k^2-2k-8 (k>=6).

Further improvements in Waring's problem

1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 2. I t e ra t ive schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3. E s t i m a t e s for the n u m b e r of so lu t ions of auxi l ia ry equa t ions . . . 155 4. M a j o r and mino r arc e s t ima tes . . . . . . . . . . . . . . . . . . . . . 168 5. T he i t e ra t ive scheme for fifth powers, I . . . . . . . . . . . . . . . . 182 6, T h e i t e ra t ive scheme for fifth powers, II . . . . . . . . . . . . . . . . 185 7. T he proof of T h e o r e m 1.1 for fifth powers . . . . . . . . . . . . . . 191 8. T he i te ra t ive scheme for k exceeding 5: second differences . . . . 197 9. T he i t e ra t ive scheme for k exceeding 5: t h i r d differences . . . . . 202 10. T he i t e ra t ive scheme for k exceeding 6: four th differences . . . . . 206 11. T h e i t e ra t ive scheme for k exceeding 7: fifth and s ix th differences 212 12. T h e proof of T h e o r e m 1.1 for s ix th powers . . . . . . . . . . . . . . 216 13. T h e H a r d y L i t t l e w o o d dissec t ion for larger k . . . . . . . . . . . . 230 14. T h e p roof of T h e o r e m 1.1 for s even th powers . . . . . . . . . . . . 232 15. T he proof of T h e o r e m 1.1 for e igh th powers . . . . . . . . . . . . . 234 16. T he proof of T h e o r e m 1.1 for n i n t h powers . . . . . . . . . . . . . . 234 Append ix . Numer ica l values of p a r a m e t e r s . . . . . . . . . . . . . . . . . 236 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

ON VINOGRADOV'S MEAN VALUE THEOREM

The object of this paper is to obtain improvements in Vinogradovs mean value theorem widely applicable in additive number theory. Let J s,k (P) denote the number of solutions of the simultaneous diophantine equations with 1 ≥ x i , y i ≥ P for 1 ≥ i ≥ s . In the mid-thirties Vinogradov developed a new method (now known as Vinogradovs mean value theorem ) which enabled him to obtain fairly strong bounds for J s,k (P) . On writing in which e (α) denotes e 2πiα , we observe that where T k denotes the k -dimensional unit cube, and α = (α 1 ,…,α k ).

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 the local solubility of diophantine systems

Let p be a rational prime number. We refine Brauers elementary diagonalisation argument to show that any system of r homogeneous polynomials of degree d, with rational coefficients, possesses a non-trivial p-adic solution provided only that the number of variables in this system exceeds (rd2)2d-1. This conclusion improves on earlier results of Leep and Schmidt, and of Schmidt. The methods extend to provide analogous conclusions in field extensions of Q, and in purely imaginary extensions of Q. We also discuss lower bounds for the number of variables required to guarantee local solubility.

Some Remarks on Vinogradov's Mean Value Theorem and Tarry's Problem

AbstractLetW(k, 2) denote the, least numbers for which the system of equations

APPROXIMATING THE MAIN CONJECTURE IN VINOGRADOV'S MEAN VALUE THEOREM

\sum\nolimits_{i = 1}^s {x_i^j = } \sum\nolimits_{i = 1}^s {y_i^j (1 \leqslant j \leqslant k)}