On the One-Sided Boundedness of Sums of Fractional Parts ({nα+γ}−image) Original Research Article

Given an irrational α in [0, 1), we ask for which values of γ in [0, 1) the sumsimageare bounded from above or from below for all m. When the partial quotients in the continued fraction expansion of α=[0, a1, a2, …] are bounded, say ailess-than-or-equals, slantA, we give a necessary condition on γ (involving the non-homogeneous continued fraction expansion of γ with respect to α). When the aigreater-or-equal, slanted2 we give examples of γ that cause one-sided boundedness. In particular, when 2less-than-or-equals, slantailess-than-or-equals, slantA and the a2i−1 (respectively a2i) are all even, we call deduce that C(m, α, γ) is bounded from below (resp. above) if and only if γ={image α+sα} (resp. γ={image α+sα}) for some integer s. The sums C(m, α, γ) are always unbounded with C(m, α, γ)>c log m for infinitely many m.