# Sums of three cubes in three linked three-progressions.

@article{Jrg1995SumsOT, title={Sums of three cubes in three linked three-progressions.}, author={Br{\"u}dern J{\"o}rg and Antal Balog}, journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)}, year={1995}, volume={1995}, pages={45 - 86} }

Let m be the set of all numbers representable äs the sum of three positive cubes. It is conjectured that # has positive density, or at least nearly so. If this were known then many interesting arithmetic properties such äs the existence of arbitrarily long arithmetic progressions would either follow from general results on dense sequences or otherwise fall within the compass of existing methods and could probably be readily established. However the density of ̂ is not known at present, the best… Expand

#### 4 Citations

ON WARING'S PROBLEM IN SUMS OF THREE CUBES

- Mathematics
- 2020

We investigate the asymptotic formula for the number of representations of a large positive integer as a sum of $k$-th powers of integers represented as the sums of three positive cubes, counted with… Expand

ON WARING'S PROBLEM FOR CUBES AND SMOOTH WEYL SUMS

- Mathematics
- 2001

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,… Expand

Diophantine approximation by cubes of primes and an almost prime. II

- Mathematics
- 2001

Let λ1, . . . , λ4 be non-zero with λ1/λ2 irrational and negative, and let S be the set of values attained by the form λ1x1 +· · ·+λ4x4 when x1 has at most 3 prime divisors and the remaining… Expand

Correlation estimates for sums of three cubes

- Mathematics
- 2014

We establish estimates for linear correlation sums involving sums of three positive integral cubes. Under appropriate conditions, the underlying methods permit us to establish the solubility of… Expand

#### References

SHOWING 1-9 OF 9 REFERENCES

A sieve approach to the Waring-Goldbach problem. I. Sums of four cubes

- Mathematics
- 1995

The problem of representing integers as the sum of k-th powers of primes is known as the Waring-Goldbach problem. Traditionally results on this problem are obtained by reference to auxiliary… Expand

On pairs of additive cubic equations.

- Mathematics
- 1988

have no nontrivial solution in the 7-adic field. However Cook [5] proved the following stronger result for primes p φ 7. Proposition l . Lei n^l3 and let p be a prime, p φ 7. Then two simultaneous… Expand

Cubic equations of additive type

- Mathematics
- Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
- 1966

It is proved that two simultaneous equations of the form axx + ...+ a nxl = 0, bxx + ... + bnxl = 0, with integral coefficients, are soluble (with not all of x1, ..., xn zero) in p-adic integers for… Expand

Brüdern, Sums of cubes of squarefree numbers

- Monatsh. Math
- 1991

Brüdern and RJ, Coak, Three additive cubic equatioas

- Acta Arith. fä
- 1991

The large sieve, Mathematika

- J. London Math. Soc
- 1936