Speaker: Demi Allen (University of Warwick)
17 December, 2021 14:30 UTC
Abstract: The classical (inhomogeneous) Khintchine-Groshev Theorem tells us that for a monotonic approximating function $\psi: \mathbb{N} \to [0,\infty)$ the Lebesgue measure of the set of (inhomogeneously) $\psi$-well-approximable points in $\mathbb{R}^{nm}$ is zero or full depending on, respectively, the convergence or divergence of $\sum_{q=1}^{\infty}{q^{n-1}\psi(q)^m}$. In the homogeneous case, it is now known that the monotonicity condition on $\psi$ can be removed whenever $nm>1$, and cannot be removed when $nm=1$. In this talk I will discuss recent work with Felipe A. RamÃrez (Wesleyan, US) in which we show that the inhomogeneous Khintchine-Groshev Theorem is true without the monotonicity assumption on $\psi$ whenever $nm>2$. This result brings the inhomogeneous theory almost in line with the completed homogeneous theory. I will survey previous results towards removing monotonicity from the homogeneous and inhomogeneous Khintchine-Groshev Theorem before discussing the main ideas behind the proof our recent result.