Smooth numbers in Beatty sequences

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

Download Full-text

Beatty sequences and Langford sequences

Discrete Mathematics ◽

10.1016/0012-365x(93)90153-k ◽

1993 ◽

Vol 111 (1-3) ◽

pp. 165-178 ◽

Cited By ~ 1

Author(s):

Roger B. Eggleton ◽

Aviezri S. Fraenkel ◽

R.Jaime Simpson

Keyword(s):

Beatty Sequences

Download Full-text

Characterization of two-distance sequences

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700035795 ◽

1992 ◽

Vol 53 (2) ◽

pp. 198-218 ◽

Cited By ~ 19

Author(s):

W. F. Lunnon ◽

P. A. B. Pleasants

Keyword(s):

Formal Languages ◽

Minimal Word ◽

Beatty Sequences ◽

Straight Lines ◽

Symbol Sequences ◽

Number Of Authors

AbstractThree differently defined classes of two-symbol sequences, which we call the two-distance sequences, the linear sequences and the characteristic sequences, have been discussed by a number of authors and some equivalences between them are known. We present a self-contained proof that the three classes are the same (when ambiguous cases of linear sequences are suitably in terpreted). Associated with each sequence is a real invariant (having a different appropriate definition for each of the three classes). We give results on the relation between sequences with the same invariant and on the symmetry of the sequences. The sequences are closely related to Beatty sequences and occur as digitized straight lines and quasicrystals. They also provide examples of minimal word proliferation in formal languages.

Download Full-text

Extended Bounds of Beatty Sequence Associated with Primes

International Journal of Engineering and Advanced Technology - Regular Issue ◽

10.35940/ijeat.f1019.0986s319 ◽

2019 ◽

Vol 8 (6S3) ◽

pp. 115-118

Keyword(s):

Character Sums ◽

Beatty Sequences ◽

Composite Moduli ◽

Beatty Sequence

This paper discusses on the estimation of character sums with respect to non-homogeneous Beatty sequences, over prime where , and is irrational. In particular, the bounds is found by extending several properties of character sums associated with composite moduli over prime. As a result, the bound of is deduced.

Download Full-text