Inhomogeneous Diophantine approximation in the coprime setting

Abstract

Given n ∈ N and x, γ ∈ R, let ||γ − nx|| ′ = min{|γ − nx + m| : m ∈ Z, gcd(n, m) = 1}, Two conjectures in the coprime inhomogeneous Diophantine approximation state that for any irrational number α and almost every γ ∈ R, lim inf n→∞ n||γ − nα|| ′ = 0 and that there exists C > 0, such that for all α ∈ R\Q and γ ∈ [0, 1) , lim inf n→∞ n||γ − nα|| ′ < C.We prove the first conjecture and disprove the second one.

On the distribution of nonprimitive lattice points in the plane

“…The proof of Theorem 1 is close to original argument from [4], as well as to the argument by Worley [5] (see also [1]).…”

“…In the present paper we improve on the result by Jitomirskaya and Liu. Combining the original argument of Theorem A by Erdős with the construction from [4] we show that there exist uncountably many α ∈ R \ Q and η ∈ R such that inf…”

“…In the last years, the problem on inhomogeneous approximation with coprime numbers again became of interest (see [4] and the bibliography therein). In particular, Jitomirskaya and Liu proved the following result.…”

On the distribution of non-primitive lattice points in the plane

“…In the present paper we improve on the result by Jitomirskaya and Liu. Combining the original argument of Theorem A by Erdős with the construction from [4] we show that there exist uncountably many α ∈ R \ Q and η ∈ R such that inf (q,r)∈Z 2 , q>100, (q,r)=1…”

Universal hierarchical structure of quasiperiodic eigenfunctions

“…Establishing this fully would require certain new ideas since so far even an arithmetic version of localization for the Diophantine case has not been established for the general analytic family, the current state-ofthe-art result by Bourgain-Goldstein [18] being measure theoretic in α. However, some ideas of our method can already be transferred to general trigonometric polynomials [35]. Moreover, our method was used recently in [27] to show that the same f and g govern the asymptotics of eigenfunctions and universality around the local maxima throughout the a.e.…”