Positive exponential sums, difference sets and recurrence

AbstractThe main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.

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Graphs of vectorial plateaued functions as difference sets

Finite Fields and Their Applications ◽

10.1016/j.ffa.2020.101795 ◽

2021 ◽

Vol 71 ◽

pp. 101795

Author(s):

Ayça Çeşmelioğlu ◽

Oktay Olmez

Keyword(s):

Difference Sets ◽

Plateaued Functions

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Divisible Semiplanes, Arcs, and Relative Difference Sets

Canadian Journal of Mathematics ◽

10.4153/cjm-1987-051-1 ◽

1987 ◽

Vol 39 (4) ◽

pp. 1001-1024 ◽

Cited By ~ 4

Author(s):

Dieter Jungnickel

Keyword(s):

Projective Plane ◽

Difference Sets ◽

Difference Set ◽

Relative Difference ◽

List Type ◽

Relative Difference Set ◽

Large Collection ◽

Baer Subplane ◽

Relative Difference Sets ◽

Definition Of

In this paper we shall be concerned with arcs of divisible semiplanes. With one exception, all known divisible semiplanes D (also called “elliptic” semiplanes) arise by omitting the empty set or a Baer subset from a projective plane Π, i.e., D = Π\S, where S is one of the following:(i) S is the empty set.(ii) S consists of a line L with all its points and a point p with all the lines through it.(iii) S is a Baer subplane of Π.We will introduce a definition of “arc” in divisible semiplanes; in the examples just mentioned, arcs of D will be arcs of Π that interact in a prescribed manner with the Baer subset S omitted. The precise definition (to be given in Section 2) is chosen in such a way that divisible semiplanes admitting an abelian Singer group (i.e., a group acting regularly on both points and lines) and then a relative difference set D will always contain a large collection of arcs related to D (to be precise, —D and all its translates will be arcs).

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On Waring’s problem: Three cubes and a sixth power

Nagoya Mathematical Journal ◽

10.1017/s0027763000007893 ◽

2001 ◽

Vol 163 ◽

pp. 13-53 ◽

Cited By ~ 4

Author(s):

Jörg Brüdern ◽

Trevor D. Wooley

Keyword(s):

Exponential Sums ◽

Large Integer ◽

Natural Numbers ◽

Waring’S Problem ◽

Smooth Numbers ◽

Waring's Problem ◽

Order Of Magnitude ◽

Almost All ◽

Hardy Littlewood Method

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.

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