Large sets in finite fields are sumsets

Alon, Noga

doi:10.1016/j.jnt.2006.11.007

Cited by 38 publications

(55 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For p = 19, by quadratic reciprocity (or directly), the degree of regularity of the graph G is r = For p = 17 the graph G is 2-regular, and is in fact a cycle: (1, 8, 13, 2, 16, 9, 4, 15) (note that all sums of adjacent elements in this cycle are quadratic residues). For p = 23 the set of quadratic residues is Q = {1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18}, the corresponding graph is 4-regular and it is not difficult to find in it a Hamilton cycle: (1, 2,16,8,4,9,3,6,18,13,12).…”

Section: Lemma 22 ([20])mentioning

confidence: 99%

See 1 more Smart Citation

Additive Patterns in Multiplicative Subgroups

Alon

Bourgain

2014

Geom. Funct. Anal.

Self Cite

View full text Add to dashboard Cite

The study of sum and product problems in finite fields motivates the investigation of additive structures in multiplicative subgroups of such fields. A simple known fact is that any multiplicative subgroup of size at least q 3/4 in the finite field F q must contain an additive relation x + y = z.Our main result is that there are infinitely many examples of sum-free multiplicative subgroups of size Ω(p 1/3 ) in prime fields F p . More complicated additive relations are studied as well. One representative result is the fact that the elements of any multiplicative subgroup H of size at least q 3/4+o(1) of F q can be arranged in a cyclic permutation so that the sum of any pair of consecutive elements in the permutation belongs to H. The proofs combine combinatorial techniques based on the spectral properties of Cayley sum-graphs with tools from algebraic and analytic number theory.

show abstract

Section: Lemma 22 ([20])mentioning

confidence: 99%

“…It is easy and well known (c.f., e.g., [2]) that the eigenvalues of G can be expressed in terms of T and the characters of B. Indeed, the eigenvalues of the square of the adjacency matrix of G are all the expressions | s∈T χ(s)| 2 , where χ is a character of B, and the characters are the corresponding eigenvectors.…”

Section: Lemma 26 ([22])mentioning

confidence: 99%

Additive Patterns in Multiplicative Subgroups

Alon

Bourgain

2014

Geom. Funct. Anal.

Self Cite

View full text Add to dashboard Cite

show abstract

“…with some absolute constants c, C > 0 for all sufficiently large p; as indicated in [1], the upper bound is likely to be close to the truth.…”

Section: Doubling the Squares (Contributed By B Green And T Tao) Hmentioning

confidence: 65%

“…Roth [75] gave an ingenious proof of this conjecture using Fourier analysis, opening the flood gates to applying Fourier methods in additive combinatorial problems. 1 At first sight, the above conjecture of Erdős and Turán may appear rather weak, for the following reason. Suppose that N is a large positive integer, and take a random integer subset A ⊆ [1, N ] with about εN elements, where ε ∈ (0, 1].…”

Section: Additive Combinatorics: a Brief Historical Overviewmentioning

confidence: 99%

Open problems in additive combinatorics

Croot¹,

Lev²

2007

CRM Proceedings and Lecture Notes

View full text Add to dashboard Cite

Abstract. A brief historical introduction to the subject of additive combinatorics and a list of challenging open problems, most of which are contributed by the leading experts in the area, are presented.In this paper we collect assorted problems in additive combinatorics, including those which we qualify as classical, those contributed by our friends and colleagues, and those raised by the present authors. The paper is organized accordingly: after a historical survey (Section 1) we pass to the classical problems (Section 2), then proceed with the contributed problems (Sections 3-6), and conclude with the original problems (Section 7). Our problem collection is somewhat eclectic and by no means pretends to be complete; the number of problems can be easily doubled or tripled. We tried to include primarily those problems we came across in our research, or at least lying close to the area of our research interests. Additive combinatorics: a brief historical overviewAs the name suggests, additive combinatorics deals with combinatorial properties of algebraic objects, typically abelian groups, rings, or fields. That is, one is interested in those combinatorial properties of the set of elements of an algebraic structure, where the corresponding algebraic operation plays a crucial role. This subject is filled with many wondrous and deep theorems; the earliest of them is, perhaps, the basic CauchyDavenport theorem, proved in 1813 by Cauchy [16] and independently rediscovered in 1935 by Davenport [24,25]. This theorem says that if p is a prime, F p denotes the finite field with p elements (notation used throughout the rest of the paper), and the subsets A, B ⊆ F p are non-empty, then the sumset A + B := {a + b : a ∈ A, b ∈ B} has at least min{p, |A| + |B| − 1} elements. The analogue of this theorem for the set Z of integers is the almost immediate assertion (left as a simple exercise to the interested reader) that |A + B| ≥ |A| + |B| − 1 holds for any finite non-empty subsets A, B ⊆ Z.The F p -version of the problem is considerably more difficult, and all presently known proofs of the Cauchy-Davenport theorem incorporate a non-trivial idea, such as the transform method (sometimes called the "intersection-union trick"), the polynomial method, or Fourier analysis. The situation becomes even more complicated when one 1 2 ERNIE CROOT AND VSEVOLOD F. LEV considers subsets of a general abelian group. An extension of the Cauchy-Davenport theorem onto this case was provided by Kneser whose celebrated result [56,57] asserts that if A and B are finite, non-empty subsets of an abelian group with |A + B| < |A| + |B| − 1, then A + B is a union of cosets of a non-zero subgroup. Further refinement of Kneser's theorem was given by Kemperman in [55].Over a century passed between Cauchy's paper [16] and the next major result in the subject, proved by Schur [83] in the early 1900's. Schur's theorem states that for every fixed integer r > 0 and every r-coloring of the set N of natural numbers, there is a monochromatic triple (x, y, z) ∈ N...

show abstract

“…The twins of the usual Cayley graphs, addition Cayley graphs (also called sum graphs) received much less attention in the literature; indeed, [1] (independence number), [2] and [11] (hamiltonicity), [3] (expander properties), and [4] (clique number) is a nearly complete list of papers, known to us, where addition Cayley graphs are addressed. To some extent, this situation may be explained by the fact that addition Cayley graphs are rather difficult to study.…”

Section: Background: Addition Cayley Graphsmentioning

confidence: 99%