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.

A sequence (x n ) on the torus is said to have Poissonian pair correlations ifIt is known that, if (x n ) has Poissonian pair correlations, then the number g(n) of different gap lengths between neighboring elements of {x 1 , . . . , x n } cannot be bounded along every index subsequence (n t ). First, we improve this by showing that the maximum among the multiplicities of the neighboring gap lengths of {x 1 , . . . , x n } is o(n), as n → ∞. Furthermore, we show that, for every function f : N + → N + with lim n f (n) = ∞, there exists a sequence (x n ) with Poissonian pair correlations and such that g(n) ≤ f (n) for all sufficiently large n. This answers negatively a question posed by G. Larcher.

We investigate the number of steps taken by three variants of the Euclidean algorithm on average over Farey fractions. We show asymptotic formulae for these averages restricted to the interval (0, 1/2), establishing that they behave differently on (0, 1/2) than they do on (1/2, 1). These results are tightly linked with the distribution of lengths of certain continued fraction expansions as well as the distribution of the involved partial quotients. As an application, we prove a conjecture of Ito on the distribution of values of Dedekind sums. The main argument is based on earlier work of Zhabitskaya, Ustinov, Bykovskiĭ and others, ultimately dating back to Lochs and Heilbronn, relating the quantities in question to counting solutions to a certain system of Diophantine inequalities. The above restriction to only half of the Farey fractions introduces additional complications.

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 [Formula: see text] is a polynomial of degree [Formula: see text] 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 [Formula: see text] belonging to a certain class of polynomials and with [Formula: see text] being an explicitly given rational function in [Formula: see text].