Small fractional parts of binary forms

Abstract

Abstract We obtain bounds on fractional parts of binary forms of the shape with $\alpha_k,\alpha_l,\ldots,\alpha_0\in{\mathbb R}$ and $l\leq k-2.$ By exploiting recent progress on Vinogradov’s mean value theorem and earlier work on exponential sums over smooth numbers, we derive estimates superior to those obtained hitherto for the best exponent σ, depending on k and $l,$ such that