On Openness of Density Points under Mappings

Abstract

Basu and Ganguly recently proved a theorem connected to the classical theorem of Steinhaus which states that $A - B$ has nonempty interior if $A$ and $B$ are Lebesgue measurable subsets of the real line, each having positive measure. The Basu and Ganguly paper deals with a particular 2-place function, namely $f(x,y) = x/y$. There is nothing special about ratios. We will extend their results to functions satisfying simple conditions on their partial derivatives. An $n$ dimensional analogue is also presented.