On ratio sets of real numbers (Q1209778)

The late H. Steinhaus proved in 1920 that whenever \(A\) and \(B\) are sets of positive Lebesgue measure then the set \(A+B\) contains an interval. The authors consider a similar type of problem for (sets \(A\) and \(B\) of positive reals and) the set \({A\over B}\). Put \(R_ 1(A,B)=\{r: \mu(\{x\in A: (\exists y\in B){x\over y}=r\})>0\}\), \(R_ 2(A,B)=\{r: \{x\in A: (\exists y\in B) {x\over y}=r\}\) is not meager\}. The results of the paper claim, e.g., that whenever \(A\) and \(B\) are of positive Lebesgue measure, then (1) the set \(R_ 1(A,B)\) contains an interval, (2) whenever \(A\) and \(B\) are of second category, having the Baire property, then \(R_ 2(A,B)\) contains an interval.