Arithmetic structure in sparse difference sets

Using a slight modification of an argument of Croot, Ruzsa and Schoen we establish a quantitative result on the existence of a dilated copy of any given configuration of integer points in sparse difference sets. More precisely, given any configuration {v₁,…,v?} of vectors in Zd, we show that if A⊂d[1,N] with |A|/Nd?CN−1/?, then there necessarily exists r≠0 such that {rv₁,…,rv?}⊆A−A.     

Optimal difference systems of sets and difference sets

Difference systems of sets (DSSs) are combinatorial structures that are generalizations of cyclic difference sets and arise in connection with code synchronization. In this paper, we give a recursive construction of DSSs with smaller redundancy from partition-type DSSs and difference sets. As applications, we obtain some new infinite classes of optimal DSSs from the known difference sets and almost difference sets.     

Partial difference sets

A (υ,k,α,β)-partial difference set in a finite group G of order υ is a subset D of G with k distinct elements such that expressions d_nd^?1₂ for d₁ and d₂ in D, represent each non-identity element not contained in D exactly α times and each non-identity element contained in D exactly α+β times. Such a set is closely related to association schemes of PBIB designs with two associate classes.     

The mann test for divisible difference sets

Generalizing a result of Jungnickel for affine difference sets, we present an existence test for arbitrary divisible difference setsD which is analogous to the well-known and powerful test of Mann for ordinary difference sets. Several applications show that this approach is of interest also in the general case.Research partially supported by NSA grant # MDA 904-87-H-2018 and by an Alexandervon-Humboldt fellowship. The author would like to thank the Mathematisches Institut der Justus-Liebig-Universität Giessen for its hospitality during the time of this research.     

Perfect difference sets constructed from Sidon sets

A set of positive integers is a perfect difference set if every nonzero integer has a unique representation as the difference of two elements of . We construct dense perfect difference sets from dense Sidon sets. As a consequence of this new approach we prove that there exists a perfect difference set such that

Affine difference sets and related factor sets

In this article we study abelian affine difference sets in connection with the related group extensions and give some results on their orders.     

Partial difference sets inp-groups

This work is partially supported by NSA grant # MDA 904-92-H-3067     

Sumsets in difference sets

We study some properties of sets of differences of dense sets in ℤ² and ℤ³ and their interplay with Bohr neighbourhoods in ℤ. We obtain, inter alia, the following results.

Special subsets of difference sets with particular emphasis on skew Hadamard difference sets

This article introduces a new approach to studying difference sets via their additive properties. We introduce the concept of special subsets, which are interesting combinatorial objects in their own right, but also provide a mechanism for measuring additive regularity. Skew Hadamard difference sets are given special attention, and the structure of their special subsets leads to several results on multipliers, including a categorisation of the full multiplier group of an abelian skew Hadamard difference set. We also count the number of ways to write elements as a product of any number of elements of a skew Hadamard difference set.      

Reversible relative difference sets

We investigate nontrivial (m, n, k, )-relative difference sets fixed by the inverse. Examples and necessary conditions on the existence of relative difference sets of this type are studied.     

More DD difference sets

Dillon and Dobbertin proved that if L := GF(2 ^m ), gcd(k, m) = 1, d := 4 ^k ? 2 ^k + 1 and Δ _k (x) := (x + 1) ^d + x ^d + 1, then B _k := L\Δ _k (L) is a difference set in the cyclic multiplicative group L  ^×  of L. Used in the proof were the auxiliary functions $c_k^{\gamma}(x) := b_k(\gamma x^{2^k+1})$ , where γ is in L  ^×  and b _k is the characteristic function of B _k on L. When m is odd $c_k^{\gamma}$ is itself the characteristic function of a cyclic difference set which is equivalent to B _k . In this paper we point out that when m is even and γ is not a cube in L then $c_k^{\gamma}$ is the characteristic function of a difference set in the elementary abelian additive group of L; i.e. $c_k^{\gamma}$ is a bent function.     

A family of difference sets

A construction is given for difference sets with parameters $\text{v =}\text{1}\text{2}\text{3}{}^{\mathrm{s+1}}\text{(3}{}^{\mathrm{s+1}}\text{? 1)}$, $\text{k =}\text{1}\text{2}\text{3}{}^{\mathrm{s}}\text{(3}{}^{\mathrm{s+1}}\text{+ 1), λ =}\text{1}\text{2}\text{3}{}^{\mathrm{s}}\text{(3}{}^{\mathrm{s}}\text{+ 1), n = 3}{}^{\mathrm{2s}}$ in certain noncyclic groups of order v. For s = 1 it is shown that the construction yields all possible difference sets with parameters (36, 15, 6, 9) in an abelian group of order 36.     

The difference between common knowledge of formulas and sets

Common knowledge can be defined in at least two ways: syntactically as the common knowledge of a set of formulas or semantically, as the meet of the knowledge partitions of the agents. In the multi-agent S5 logic with either finitely or countably many agents and primitive propositions, the semantic definition is the finer one. For every subset of formulas that can be held in common knowledge, there is either only one member or uncountably many members of the meet partition with this subset of formulas held in common knowledge. If there are at least two agents, there are uncountably many members of the meet partition where only the tautologies of the multi-agent S5 logic are held in common knowledge. Whether or not a member of the meet partition is the only one corresponding to a set of formulas held in common knowledge has radical implications for its topological and combinatorial structure.