A result of H. Steinhaus states that any Lebesgue measurable set $X\subset R$ with positive Lebesgue measure has a property that its difference set contains an open interval around zero. In this note we will prove a statement, which, in a sense, complements it.
"A converse to a theorem of Steinhaus." Real Anal. Exchange 31 (1) 291 - 294, 2005-2006.