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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 741.52010

**Autor: ** Erdös, Paul; Makai, Endre; Pach, János; Spencer, Joel

**Title: ** Gaps in difference sets, and the graph of nearly equal distances. (In English)

**Source: ** Applied geometry and discrete mathematics, Festschr. 65th Birthday Victor Klee, DIMACS, Ser. Discret. Math. Theor. Comput. Sci. 4, 265-273 (1991).

**Review: ** [For the entire collection see Zbl 726.00015.]

Given n real numbers mutually differing by at least 1, let 1 \leq d_{1} \leq ... \leq d_{m}, m = \binom{n}{2}, denote the increasing sequence of differences between them. The authors prove the asymptotically sharp bound **sum**_{j = 1}^{m-1}(d_{j+1}-d_{j})^{2} > c· log n, and give sharp lower bounds for **sum**_{j = 1}^{m-1}\phi(d_{j+1}-d_{j}) for large class of monotone increasing convex functions \phi. They also show that for n points in the plane with minimal distance 1 and any real number t at most \lceil n^{2}/4\rceil may have distances between t and t+1, and give an analogous result in terms of the diameter of the point set.

**Reviewer: ** C.Schulz (Wiesbaden)

**Classif.: ** * 52A37 Other problems of combinatorial convexity

05B10 Difference sets

52C10 Erdoes problems and related topics of discrete geometry

**Keywords: ** difference set; distance set

**Citations: ** Zbl 726.00015

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