On small fractional parts of polynomial-like functions

In a recent paper, Madritsch and Tichy established Diophantine inequalities for the fractional parts of polynomial-like functions. In particular, for f ( x ) = x k + x c where k is a positive integer, c > 1 is a non-integer and ξ a real number, they obtained min 2 ≤ p ≤ X ‖ ξ ⌊ f ( p ) ⌋ ‖ ≪ k , c , ϵ X - ρ 2 ( c , k ) + ϵ for ρ 2 ( c , k ) > 0 explicitly given. In the present note, we improve upon their results of the case c > k and c > 4 .