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**Zentralblatt MATH**

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

**Zbl.No: ** 822.52004

**Autor: ** Elekes, G.; Erdös, Paul

**Title: ** Similar configurations and pseudo grids. (In English)

**Source: ** Böröczky, K. (ed.) et al., Intuitive geometry. Proceedings of the 3rd international conference held in Szeged, Hungary, from 2 to 7 September, 1991. Amsterdam: North-Holland, Colloq. Math. Soc. János Bolyai. 63, 85-104 (1994).

**Review: ** Given a finite set *P* of points in the plane, it is of interest to find another, n-point, set *A* which contains many similar or homothetic copies of *P*. When *P* is a square or equilateral triangle, it is easily seen that a suitable regular grid achieves c · n^{2} similar copies and c · n^{3/2} homothetic copies.

Such constructions might appear to depend on the fact that the square and equilateral triangle tile the plane in a lattice tiling. The authors show that this is not the case. If *P* is a triangle, or can be represented in the complex plane by a set of algebraic complex numbers **{**\xi_{i}**}**, then there exist ``pseudogrids'' containing c · n^{2} similar copies or c · n^{3/2} homothetic copies. The points of these pseudogrids are the values of polynomials in **{**\xi_{i}**}** with certain bounds on coefficients and degree. Even if *P* cannot be so represented, O(n^{2}) similar copies and O(n^{3/2}) homothetic copies may be achieved.

The results for homothetic copies are extended to higher dimensions. If R^{d} it is possible to construct pseudogrids which contain c · n^{(d+1)/d} homothetic copies of a pattern *P* with algebraic coordinates, or O(n^{(d+1)/d}) homothetic copies of an arbitrary pattern.

**Reviewer: ** R.Dawson (Halifax)

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

52C20 Tilings in 2 dimensions (discrete geometry)

**Keywords: ** similar configurations; homothetic configurations; pseudogrids

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag