A kronecker-weyl theorem modulo 2.

0, obtained from the continued fraction representation 0 = [0; a,, a2, .. J.. We recall that 0 is said to have bounded partial quotients when sup a; < co. I THEOREM 1. A necessary and sufficient condition for i'e(I) to exist for arbitrary I is that 0 have bounded partial quotients. Let f be the function which is -1 on I and 1 on the complement of I. Using an idea of Furstenberg's,2 one sees the existence of equation (1) as being related to the nonexistence of a measurable, ± 1-valued solution to