A visit to the Erdös problem

Abstract

Erdős asked if for every infinite set, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, of real numbers there exists a measurable subset of the reals having positive measure that does not contain a subset similar to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this note we transform this question to a finite combinatorial problem. Using this translation we extend some results of Eigen and Falconer concerning slow sequences for which the answer to Erdős’ question is positive.