Report on some recent advances in Diophantine approximation

定量丢番图逼近度量理论中的有效结果

Effective Results in The Metric Theory of Quantitative Diophantine Approximation

本文将丢番图逼近度量理论中依赖隐式常数的渐近定量化为有效结果，在测度至多 δ 的例外集外显式控制常数，并推出 Schmidt 定理、Koukoulopoulos–Maynard 定理及正规数等结果。

Diophantine Approximation and applications in Interference Alignment

This paper is motivated by recent applications of Diophantine approximation in electronics, in particular, in the rapidly developing area of Interference Alignment. Some remarkable advances in this area give substantial credit to the fundamental Khintchine-Groshev Theorem and, in particular, to its far reaching generalisation for submanifolds of a Euclidean space. With a view towards the aforementioned applications, here we introduce and prove quantitative explicit generalisations of the Khintchine-Groshev Theorem for non-degenerate submanifolds of $\mathbb{R}^n$. The importance of such quanti…

Jacobi sums and cyclotomic numbers: A survey report

The determination of Jacobi sums, their congruences and cyclotomic numbers have been the object of attention for many years and there are large number of interesting results related to these in the literature. This survey aims at reviewing results concerning the diophantine systems for finding the cyclotomic numbers and coefficients of Jacobi sums and to indicate the current status of the problem.

Counter-examples in Parametric Geometry of Numbers

Thanks to recent advances in parametric geometry of numbers, we know that the spectrum of any set of $m$ exponents of Diophantine approximation to points in $\mathbb{R}^n$ (in a general abstract setting) is a compact connected subset of $\mathbb{R}^m$. Moreover, this set is semi-algebraic and closed under coordinate-wise minimum for $n\le 3$. In this paper, we give examples showing that for $n\ge 4$ each of the latter properties may fail.

等价极小模型程序的进展及其在复动力学和算术动力学中的应用

Advances in the equivariant minimal model program and their applications in complex and arithmetic dynamics

本文探讨了非同构可缩映射的等价极小模型程序进展及其在复动力学和算术动力学中的应用。