LARGE SETS AVOIDING INFINITE ARITHMETIC / GEOMETRIC PROGRESSIONS

We study some variants of the Erd.os similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset E of the real line such that 0 is a Lebesgue density point of E, but E does not contain any (non-constant) infinite geometric progression. We give a sufficient density type condition that guarantees that a set contains an infinite geometric progression. By slightly improving a recent result of Bradford, Kohut and Mooroogen we construct a closed set F subset of [0,infinity) such that the measure of F boolean AND [t, t + 1] tends to 1 at infinity but F does not contain any infinite arithmetic progression. We also slightly improve a more general recent result by Kolountzakis and Papageorgiou for more general sequences. We give a sufficient condition that guarantees that a given Cantor type set contains at least one infinite geometric progression with any quotient between 0 and 1. This can be applied to most symmetric Cantor sets of positive measure.