On small fractional parts of perturbed polynomials

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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 [Formula: see text] where P is a polynomial of degree k and [Formula: see text] is a linear combination of functions of shape [Formula: see text], [Formula: see text], [Formula: see text]. We prove that for any given irrational [Formula: see text] we have [Formula: see text] for P belonging to a certain class of polynomials and with [Formula: see text] being an explicitly given rational function in k.