On the one-sided boundedness of sums of fractional parts ({nα+γ}-1/2)

Given an irrational alpha in [0, 1), we ask for which values of gamma in [0, 1) the sums [GRAPHICS] are bounded from above or from below for all m. When the partial quotients in the continued fraction expansion of alpha = [0, a(1), a(2), ...] are bounded, say a(i) less than or equal to A, we give a necessary condition on gamma (involving the non-homogeneous continued fraction expansion of gamma with respect to alpha. When the a(i) greater than or equal to 2 we give examples of y that cause one-sided boundedness. In particular, when 2 less than or equal to a(i) less than or equal to A and the a(2i-l) (respectively a(2i)) are all even, we call deduce that C(m, alpha, gamma) is bounded from below (resp. above) if and only if gamma = {1/2 alpha + s alpha} resp. gamma = {1/2 alpha + s alpha}) for some integer s. The sums [C(m, alpha, gamma)] are always unbounded with [C(m, alpha, gamma)] > c log m for infinitely many m. (C) 2000 Academic Press.