On small fractional parts of polynomial-like functions

Abstract

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 and c > 1 is a non-integer, and any fixed ξ ∈ [0, 1] they obtainedfor ρ 1 (c, k) > 0 explicitly given. In the present note, we improve upon their results in the case c > k and c > 4.

On small fractional parts of perturbed polynomials

“…Goal of the present note. In an earlier work (see [13]) we improved the results in [12] for the case of dominant f . More specifically, we obtained Theorem 1.2.…”

“…The proof of the latter lemma follows mutatis mutandis through the same standard lines of [13]. Nevertheless, we shall give the details for completeness.…”

“…We remark that for technical reasons, the case under consideration requires more work than in [13]. In particular, the estimation of Type I and Type II sums we will require in this work are slightly more intricate and need a further assumption on the structure of the polynomial.…”

“…The proof of the exponential sums bounds is the most technical part, and is deferred to Section 4, Section 5 and Section 6. The strategy of proof is similar to the one employed in [12] and [13]. Most of our efforts will be spent in the last two sections for obtaining bounds for prime exponential sums.…”