On discrepancy, intrinsic Diophantine approximation, and spectral gaps

On the metric theory of inhomogeneous Diophantine approximation: An Erdős-Vaaler type result

In 1958, Szüsz proved an inhomogeneous version of Khintchine's theorem on Diophantine approximation. Szüsz's theorem states that for any non-increasing approximation function $ψ:\mathbb{N}\to (0,1/2)$ with $\sum_q ψ(q)=\infty$ and any number $γ,$ the following set \[ W(ψ,γ)=\{x\in [0,1]: |qx-p-γ|< ψ(q) \text{ for infinitely many } q,p\in\mathbb{N}\} \] has full Lebesgue measure. Since then, there are very few results in relaxing the monotonicity condition. In this paper, we show that if $γ$ is can not be approximate by rational numbers too well, then the monotonicity condition can be replaced …

On the discrepancy between best and uniform approximation

For $ζ$ a transcendental real number, we consider the classical Diophantine exponents $w_{n}(ζ)$ and $\widehat{w}_{n}(ζ)$. They measure how small $| P(ζ)|$ can be for an integer polynomial $P$ of degree at most $n$ and naive height bounded by $X$, for arbitrarily large and all large $X$, respectively. The discrepancy between the exponents $w_{n}(ζ)$ and $\widehat{w}_{n}(ζ)$ has attracted interest recently. Studying parametric geometry of numbers, W. Schmidt and L. Summerer were the first to refine the trivial inequality $w_{n}(ζ)\geq \widehat{w}_{n}(ζ)$. Y. Bugeaud and the author found another…

三间隙定理的频率问题

The frequency problem of the three gap theorem

本文通过cake model研究三间隙定理中不同间隙出现的频率，证明在Lebesgue测度意义下几乎所有无理数α的三种间隙出现频率为1，两种间隙频率为0。

Gaps problems and frequencies of patches in cut and project sets

We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets o…

Higher dimensional Steinhaus and Slater problems via homogeneous dynamics

The three gap theorem, also known as the Steinhaus conjecture or three distance theorem, states that the gaps in the fractional parts of $α,2α,\ldots, Nα$ take at most three distinct values. Motivated by a question of Erdős, Geelen and Simpson, we explore a higher-dimensional variant, which asks for the number of gaps between the fractional parts of a linear form. Using the ergodic properties of the diagonal action on the space of lattices, we prove that for almost all parameter values the number of distinct gaps in the higher dimensional problem is unbounded. Our results in particular improve…

Random Walks, Spectral Gaps, and Khintchine's Theorem on Fractals

This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle $1/3$ set. We obtain the first instances where a complete analogue of Khintchine's Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of $\mathbb{R}^d$ (for any $d\geq 1$) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler's problem is the Hausdorff measure on the "middle $1/5$ Cantor set"; i.e. the set o…

The Robin problem on rectangles

We study the statistics and the arithmetic properties of the Robin spectrum of a rectangle. A number of results are obtained for the multiplicities in the spectrum, depending on the Diophantine nature of the aspect ratio. In particular, it is shown that for the square, unlike the case of Neumann eigenvalues where there are unbounded multiplicities of arithmetic origin, there are no multiplicities in the Robin spectrum for sufficiently small (but nonzero) Robin parameter except a systematic symmetry. In addition, uniform lower and upper bounds are established for the Robin-Neumann gaps in terms…

圆上线性形式偏差的几乎必然界

Almost Sure Bounds for Discrepancies of Linear Forms on the Circle

研究圆上线性形式序列的偏差，证明对于几乎全部向量，偏差主次阶为(log N)^d，改进了Beck关于多维Kronecker序列的定理。

关于与Wirsing问题相关的两个一致逼近指数

On two uniform exponents of approximation related to Wirsing's problem

本研究填补了Bugeaud提出的两个一致丢番图逼近指数不等式的证明空白，在低维等情形下验证了该不等式，还给出其推广形式的反例，推进了对Wirsing问题相关核心差异的理解。