On small fractional parts of perturbed polynomials

Questions concerning small fractional parts of polynomials and pseudo-polynomials have a long history in analytic number theory. In this paper, we improve on the earlier work by Madritsch and Tichy. In particular, let f = P + φ where P is a polynomial of degree k and φ is a linear combination of functions of shape xc, c, 1 < c < k. We prove that for any given irrational ζ we have min 2 ≤ p ≤ Xpprime ζf(p)f,X-ρ(k)+, for P belonging to a certain class of polynomials and with ρ(k) > 0 being an explicitly given rational function in k.