2018/09/02 by Glasscock, Daniel
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1809.00360
Our main result concerns a perturbation of a classic theorem of Khintchine in Diophantine approximation. We give sufficient conditions on a sequence of positive real numbers (ψn)n ∈ ℕ and differentiable functions (φn: J → ℝ)n ∈ ℕ so that for Lebesgue-a.e. θ∈ J, the inequality ‖ nθ+ φn(θ) ‖ ≤ ψn has infinitely many solutions. The main novelty is that the magnitude of the perturbation |φn(θ)| is allowed to exceed ψn, changing the usual "shrinking targets" problem into a "shifting targets" problem. As an application of the main result, we prove that if the linear equation y=ax+b, a, b ∈ ℝ, has infinitely many solutions in ℕ, then for Lebesgue-a.e. α> 1, it has infinitely many or finitely many solutions of the form \lfloor nα\rfloor according as α< 2 or α> 2.