Sum and difference free sets.

Abstract

In this paper we prove that if $X$ is an uncountable subset of the reals and $\kappa $ is a cardinal smaller than the cardinality of the set $X $ then the algebraic difference $X-X$ of the set $X$ is not a finite union of $\kappa $ sum free or $\kappa$ difference free sets. An application of the above result is that for any function $f:\mathbb{R}\to \{1,2,...,n\} $ and for each cardinal $\lambda <2^{\omega } $the set of all $x $ such that ${|\{h>0:f(x-h)=f(x+h)\}|\geq \lambda } $ is of the size of the continuum. Among other things, we show that a finite union of countably many translates of $2^{\omega } $ difference free subsets of the reals is not residual in an interval. In the above statement, \textquotedblleft countably many\textquotedblright can be replaced by "fewer than continuum many" provided that $2^{\omega } $ is a regular cardinal