Progress on local properties problems of difference sets

Embeddability Properties of Difference Sets

By using nonstandard analysis, we prove embeddability properties of di↵erences A B of sets of integers. (A set A is “embeddable” into B if every finite configuration of A has shifted copies in B.) As corollaries of our main theorem, we obtain improvements of results by I.Z. Ruzsa about intersections of di↵erence sets, and of Jin’s theorem (as refined by V. Bergelson, H. Fürstenberg and B. Weiss...

A Brief Survey of Difference Sets, Partial Difference Sets, and Relative Difference Sets

Difference sets, partial difference sets, and relative difference sets are algebraic structures with combinatorial applications. The purpose of this paper is to give the reader definitions and basic examples of these sets, as well describe their connections to some other well-known combinatorial objects. We will conclude with some examples that illustrate how character theory can be used to pro...

Focusing on Local Difference

The external granular layer (EGL) of the cerebellum thickens, thins and disappears in its developmental process. We examined the thickness of the EGL, both intralobule differences and interlobule differences, in the whole midline sections of mouse cerebella for the entire postnatal period up to disappearance. The thinnest site in each lobule was located at the outer apex throughout the observat...

Extremal problems for local properties of graphs

On some new constructions of difference sets

Difference sets are mathematical structures which arise in algebra and combinatorics, with applications in coding theory. The fundamental question is when and how one can construct difference sets. This largely expository paper looks at standard construction methods and describes recent findings that resulted in new families of difference sets. This paper provides explicit examples of differenc...