We generalize a classical Steinhaus theorem replacing addition by any two variable function which is differentiable. We deal with category and measure case. Measure case was done before in papers , . We give a proof which can be easily generalized to category case. We also obtain a variety of examples of nonmeasurable sets which have interesting algebraic properties.
"A Generalization of Steinhaus' Theorem and Some Nonmeasurable Sets." Real Anal. Exchange 35 (2) 403 - 412, 2009/2010.