Davenport Theorem Research Articles

A Cauchy–Davenport Theorem for Locally Compact Groups

Abstract We generalize the Cauchy–Davenport theorem to all locally compact groups.

An Elementary Method in Number Theory

In order to apply the role of the analogues between integers and polynomials, in this paper we introduce a unified diagram to the Fermat's theorem, Mason's theorem, Davenport's theorem and some arithmetic conjectures. Further, we show some applications of this diagram in teaching and researching on arithmetic. By using this elementary method in number theory, we obtained some results for polynomials and holomorphic functions in case complex and p-adic.

Some Generalizations of the Cauchy–Davenport Theorem

We investigate two possible generalizations of the Cauchy–Davenport inequality |A + B| ≥ min (p, |A| + |B| − 1) for nonempty sets A, B of residues modulo a prime number p. The first one deals with another way of measuring the size of a set of points in an affine space (rather than just taking the cardinality), namely, with algebraic complexity. The second one concentrates on the multiplicative group of a field.

Davenport's theorem in geometric discrepancy theory

Davenport's theorem was established nearly a lifetime ago, but there have been some very interesting recent developments. The various proofs over the years bring in different ideas from number theory, probability theory, analysis and group theory. In this short survey, we shall not present complete proofs, but will describe instead some of these underlying ideas.

CAUCHY–DAVENPORT TYPE THEOREMS FOR SEMIGROUPS

Let be a (possibly non-commutative) semigroup. For , we define , where is the set of the units of and The paper investigates some properties of and shows the following extension of the Cauchy–Davenport theorem: if is cancellative and , then This implies a generalization of Kemperman's inequality for torsion-free groups and strengthens another extension of the Cauchy–Davenport theorem, where is a group and in the above is replaced by the infimum of as ranges over the non-trivial subgroups of (Hamidoune–Károlyi theorem).

Extending Brickell–Davenport theorem to non-perfect secret sharing schemes

One important result in secret sharing is the Brickell---Davenport theorem: every ideal perfect secret sharing scheme defines a matroid that is uniquely determined by the access structure. We present a generalization of the Brickell---Davenport theorem to the general case, in which non-perfect schemes are also considered. After analyzing that result under a new point of view and identifying its combinatorial nature, we present a characterization of the (not necessarily perfect) secret sharing schemes that are associated with matroids. Some optimality properties of such schemes are discussed.

Note on a conjecture of Graham

An old conjecture of Graham stated that if n is a prime and S is a sequence of n terms from the cyclic group Cn such that all (nontrivial) zero-sum subsequences have the same length, then S must contain at most two distinct terms. In 1976, Erdős and Szemerédi gave a proof of the conjecture for sufficiently large primes n. However, the proof was complicated enough that the details for small primes were never worked out. Both in the paper of Erdős and Szemerédi and in a later survey by Erdős and Graham, the complexity of the proof was lamented. Recently, a new proof, valid even for non-primes n, was given by Gao, Hamidoune and Wang, using Savchev and Chen’s recently proved structure theorem for zero-sum free sequences of long length in Cn. However, as this is a fairly involved result, they did not believe it to be the simple proof sought by Erdős, Graham and Szemerédi. In this paper, we give a short proof of the original conjecture that uses only the Cauchy–Davenport Theorem and pigeonhole principle, thus perhaps qualifying as a simple proof. Replacing the use of the Cauchy–Davenport Theorem with the Devos–Goddyn–Mohar Theorem, we obtain an alternate proof, albeit not as simple, of the non-prime case. Additionally, our method yields an exhaustive list detailing the precise structure of S and works for an arbitrary finite abelian group, though the only non-cyclic group for which the hypotheses are non-void is C2⊕C2m.

A Cauchy–Davenport Type Result for Arbitrary Regular Graphs

AbstractMotivated by the Cauchy–Davenport theorem for sumsets, and its interpretation in terms of Cayley graphs, we prove the following main result: There is a universal constant

A property of groups and the Cauchy–Davenport theorem

We introduce and study a property of groups which could be regarded as a broad generalization of the Cauchy–Davenport theorem and which turns out to be useful in investigation of the intersection of subgroups in free products of groups.

The extent to which subsets are additively closed

Given a finite abelian group G (written additively), and a subset S of G, the size r ( S ) of the set { ( a , b ) : a , b , a + b ∈ S } may range between 0 and | S | 2 , with the extremal values of r ( S ) corresponding to sum-free subsets and subgroups of G. In this paper, we consider the intermediate values which r ( S ) may take, particularly in the setting where G is Z / p Z under addition ( p prime). We obtain various bounds and results. In the Z / p Z setting, this work may be viewed as a subset generalization of the Cauchy–Davenport Theorem.

A variant of Tao's method with application to restricted sumsets

In this paper, we develop Terence Tao's harmonic analysis method and apply it to restricted sumsets. The well-known Cauchy–Davenport theorem asserts that if ∅ ≠ A , B ⊆ Z / p Z with p a prime, then | A + B | ⩾ min { p , | A | + | B | − 1 } , where A + B = { a + b : a ∈ A , b ∈ B } . In 2005, Terence Tao gave a harmonic analysis proof of the Cauchy–Davenport theorem, by applying a new form of the uncertainty principle on Fourier transform. We modify Tao's method so that it can be used to prove the following extension of the Erdős–Heilbronn conjecture: If A , B , S are non-empty subsets of Z / p Z with p a prime, then | { a + b : a ∈ A , b ∈ B , a − b ∉ S } | ⩾ min { p , | A | + | B | − 2 | S | − 1 } .

Some results on minimal sumset sizes in finite non-abelian groups

Let G be a group. We study the minimal sumset (or product set) size μ G ( r , s ) = min { | A ⋅ B | } , where A , B range over all subsets of G with cardinality r , s respectively. The function μ G has recently been fully determined in [S. Eliahou, M. Kervaire, A. Plagne, Optimally small sumsets in finite abelian groups, J. Number Theory 101 (2003) 338–348; S. Eliahou, M. Kervaire, Minimal sumsets in infinite abelian groups, J. Algebra 287 (2005) 449–457] for G abelian. Here we focus on the largely open case where G is finite non-abelian. We obtain results on μ G ( r , s ) in certain ranges for r and s, for instance when r ⩽ 3 or when r + s ⩾ | G | − 1 , and under some more technical conditions. (See Theorem 4.4.) We also compute μ G for a few non-abelian groups of small order. These results extend the Cauchy–Davenport theorem, which determines μ G ( r , s ) for G a cyclic group of prime order.

On a partition analog of the Cauchy-Davenport theorem

Let G be a finite abelian group, and let n be a positive integer. From the Cauchy-Davenport Theorem it follows that if G is a cyclic group of prime order, then any collection of n subsets A_1,A_2,\ldots,A_n of G satisfies \bigg|\sum_{i=1}^n A_i\bigg| \ge \min \bigg\{|G|,\,\sum_{i=1}^n |A_i|-n+1\bigg\}. M.~Kneser generalized the Cauchy--Davenport Theorem for any abelian group. In this paper, we prove a sequence-partition analog of the Cauchy--Davenport Theorem along the lines of Kneser's Theorem. A particular case of our theorem was proved by J.~E. Olson in the context of the Erdos--Ginzburg--Ziv Theorem.

Value sets of polynomials and the Cauchy Davenport theorem

We define an invariant for any finite sequence of elements belonging to a field. We find a lower bound for the cardinality of the underlying set of distinct elements of the finite sequence in terms of this invariant. We use this bound to study sumsets of finite subsets of a given field. The motivation for our method comes from a lower bound on the value set of a polynomial over a finite field, considered by Wan et al. We give a new proof of the Cauchy Davenport theorem and show how it may be applied in the case of prime power fields. We apply our results to study value sets of diagonal polynomials over finite fields. Our methods may have applications to Waring's problem over prime power fields. This calls for further investigations.

Davenport's theorem in the theory of irregularities of point distribution

We study distributions \(\mathcal{D}_N \) of N points in the unit square U2 with minimal order of L2-discrepancy ℒ2[\(\mathcal{D}_N \)] < C(log N)1/2, where the constant C is independent of N. We present an approach that uses Walsh functions and can be generalized to higher dimensions. Bibliography: 19 titles.

Generalized derivations and additive theory

In this paper we investigate cyclic spaces of generalized derivations related to the symmetric functions, and its relation with a generalization of the Cauchy–Davenport Theorem.

The invariant polynomials degrees of the Kronecker sum of two linear operators and additive theory

Let G be an abelian group. Let A and B be finite non-empty subsets of G. By A+B we denote the set of all elements a+b with a∈A and b∈B. For c∈A+B, ν c(A,B) is the cardinality of the set of pairs (a,b) such that a+b=c. We call ν c(A,B) the multiplicity of c ( in A+B). Let i be a positive integer. We denote by μ i(A,B) or briefly by μ i the cardinality of the set of the elements of A+B that have multiplicity greater than or equal to i. Let F be a field. Let p be the characteristic of F in case of finite characteristic and ∞ if F has characteristic 0. Let A and B be finite non-empty subsets of F . We will prove that for every ℓ=1,…, min{|A|,|B|} one has (a) μ 1+⋯+μ ℓ⩾ℓ min{p,|A|+|B|−ℓ}. This statement on the multiplicities of the elements of A+B generalizes Cauchy–Davenport Theorem. In fact Cauchy–Davenport is exactly inequality (a) for ℓ=1. When F= Z p inequality (a) was proved in J.M. Pollard (J. London Math. Soc. 8 (1974) 460–462); see also M.B. Nathanson (Additive number theory: Inverse problems and the geometry of sumsets, Springer, New York, 1996).

Sumsets in Vector Spaces over Finite Fields

We determine explicitly the least possible size of the sumset of two subsetsA, B⊂(Z/pZ)Nwith fixed cardinalities, thereby generalizing both Cauchy–Davenport's theorem (caseN=1) and Yuzvinsky's theorem(casep=2). The solution involves a natural generalization of the well-known Hopf–Stiefel–Pfister function. The corresponding problem for more than two summands is also considered and solved. We then consider restricted sumsets, formed by taking sums of distinct elements only. We determine almost completely the least possible size of the restricted sumset of two subsets in (Z/pZ)Nwith fixed cardinalities. Our result generalizes the recent solution(s) of the Erdős–Heilbronn conjecture dealing with the restricted sumsets of two equal subsets inZ/pZ.

The Polynomial Method and Restricted Sums of Congruence Classes

We present a simple and general algebraic technique for obtaining results in Additive Number Theory, and apply it to derive various new extensions of the Cauchy–Davenport Theorem. In particular we obtain, for subsetsA0, A1, …, Akof the finite fieldZp, a tight lower bound on the minimum possible cardinality of[formula]as a function of the cardinalities of the setsAi.

On Bombieri and Davenport's theorem concerning small gaps between primes

MathematikaVolume 39, Issue 1 p. 10-17 Research Article On Bombieri and Davenport's theorem concerning small gaps between primes D. A. Goldston, D. A. Goldston Department of Mathematics and Computer Science, San Jose State University, San Jose, CA, 95192 USASearch for more papers by this author D. A. Goldston, D. A. Goldston Department of Mathematics and Computer Science, San Jose State University, San Jose, CA, 95192 USASearch for more papers by this author First published: 01 June 1992 https://doi.org/10.1112/S0025579300006793Citations: 7AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume39, Issue1June 1992Pages 10-17 RelatedInformation