In 1960, he proved that every normal number in base r is normal in base s if and only if log r / log s is a rational number. He also proved the existence of T numbers. His series of papers on irregularities of distribution can be seen in J.Beck and W.Chen, Irregularities of Distribution, Cambridge University Press.
Schmidt is in a small group of number theorists who have been invited to address the International Congress of Mathematicians three times. The others are Iwaniec, Shimura, and Tate.