Martin Lind
We consider a certain equidistributed sequence of rational numbers constructed from the primes. In particular, we determine the sharp convergence rate for the star discrepancy of said sequence. Our arguments are based on well-known discrepancy estimates for inversive congruential pseudorandom numbers together with asymptotic formulae involving prime numbers.
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We consider the star discrepancy of two-dimensional sequences made up as a hybrid between a Kronecker sequence and a perturbed Halton sequence in base 2, where the perturbation is achieved by a digital-sequence construction in the sense of Niederreiter whose generating matrix contains a periodic perturbing sequence of a given period length. Under the assumption that the Kronecker sequence involves a parameter with bounded continued fraction coefficients sharp discrepancy estimates are obtained. Furthermore, we study the problem from a metric point of view as well. Finally, we also present shar…
The lattice discrepancy of certain three-dimensional bodies
基于一个相当精确的近似的拉梅盘的晶格差异，一个渐近公式建立了一个相关的三维体的晶格点的数量，线性膨胀的一个大的实参数x。特别注意高斯曲率为零的边界点。
Let $(n_k)_{k=1}^{\infty}$ be a lacunary sequence of integers. We show that if $μ$ is a probability measure on $[0,1)$ such that $|\widehatμ(t)|\leq c|t|^{-η}$, then for $μ$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies \begin{equation*} \frac{1}{4} \leq \limsup_{N\to\infty}\frac{N D_N(n_kx)}{\sqrt{N\log\log N}} \leq C \end{equation*} for some constant $C>0$, proving a conjecture of Haynes, Jensen and Kristensen. This allows a slight improvement on their previous result on products of the form $q\|qα\| \|qβ-γ\| $.
Discrepancy estimates for sequences: new results and open problems
在本文中，我们给出了一个概述最近的结果（上，下）的差异估计（具体）序列的单位立方体。特别是，我们也给出了某些类别的混合序列的差异估计的概述。这里，通过混合序列，我们理解为$(s+t)$维序列，其是特定类型的$s$维序列（例如Kronecker-、Niederreiter-、Halton-、.类型）和另一类型的$t$维序列。杂交序列（及其组成部分）的差异分析是一个相当流行和生动的研究分支。我们收集了一些具有挑战性的开放问题在这个主题上。
Sharp density discrepancy for cut and project sets: An approach via lattice point counting
本文研究切割与投影集的点计数渐近波动，获得了依赖于格丢番图性质的偏差一致上下界，并证明了其尖锐性，同时找到了与格双Lipschitz等价的最大切割与投影集类。
The discrepancy function measures the deviation of the empirical distribution of a point set in $[0,1]^d$ from the uniform distribution. In this paper, we study the classical discrepancy function with respect to the BMO and exponential Orlicz norms, as well as Sobolev, Besov and Triebel-Lizorkin norms with dominating mixed smoothness. We give sharp bounds for the discrepancy function under such norms with respect to infinite sequences.
Mean lattice point discrepancy bounds, II: Convex domains in the plane
我们考虑平面弯曲严格凸区域没有或非常弱的光滑性假设，并证明了与格点差异相关的平方函数的严格界。
We prove sharp estimates for Fourier transforms of indicator functions of bounded open sets in ${\mathbb R}^n$ with real analytic boundary, as well as nontrivial lattice point discrepancy results. Both will be derived from estimates on Fourier transforms of hypersurface measures. Relations with maximal averages are discussed, connecting two conjectures of Iosevich and Sawyer from [ISa1]. We also prove a theorem concerning the stability under function perturbations of the growth rate of a real analytic function near a zero. This result is sharp in an appropriate sense. It implies a correspondin…
We study the $L_p$ discrepancy of digital NUT sequences which are an important sub-class of digital $(0,1)$-sequences in the sense of Niederreiter. The main result is a lower bound for certain sub-classes of digital NUT sequences.
We consider random lattices taken from the general symplectic ensemble and count the number of lattice points of a typical lattice in nested families $B_t$ of certain Borel sets. Our main result is that for almost every general symplectic lattice, the discrepancy $D(Λ,B_t)$ of the lattice point count with respect to the volumes is $O(vol(B_t)^{-δ})$. This extends work of W. Schmidt who gave similar discrepancy bounds for the space of all lattices in $\mathbb{R}^n$.
The goal of this overview article is to give a tangible presentation of recent breakthrough works in discrepancy theory by M. B. Levin. These works provide proofs for the exact lower discrepancy bounds of Halton's sequence and a certain class of $(t, s)$-sequences. Our survey aims at highlighting the major ideas of the proofs and we discuss further implications of the employed methods. Moreover, we derive extensions of Levin's results.
Semi-algebraic discrepancy estimates for multi-frequency shift sequences with applications to quantum dynamics
本文研究多频平移序列的差异估计，建立渐近精确的半代数界，并将其用于推导长程拟周期 Schrödinger 算子量子动力学的新上界。
Estimates of the Discrepancy Function in Exponential Orlicz Spaces
本文证明了在n ≥ 3的所有维数上，对于任意整数N，存在基数点分布$ N$，其相关的偏差函数D_N满足指数Orlicz类exp）L^{2/（n+1）}中N的急剧增长率的估计.这一点最近已被M.~证明。Skriganov，使用二进制数字网的随机数字移位，建立在W.L.~ Chen和M.~斯克里加诺夫我们独立开发的方法是对Skriganov方法的补充。
Let A_N be an N-point distribution in the unit square in the Euclidean plane. We consider the Discrepancy function D_N(x) in two dimensions with respect to rectangles with lower left corner anchored at the origin and upper right corner at the point x. This is the difference between the actual number of points of A_N in such a rectangle and the expected number of points - N x_1x_2 - in the rectangle. We prove sharp estimates for the BMO norm and the exponential squared Orlicz norm of D_N(x). For example we show that necessarily ||D_N||_(expL^2) >c(logN)^(1/2) for some aboslute constant c>0. On …
We say that a sequence $\{x_n\}_{n \geq 1}$ in $[0,1)$ has Poissonian pair correlations if \begin{equation*} \lim_{N \rightarrow \infty} \frac{1}{N} \# \left\{ 1 \leq l \neq m \leq N \, : \, \left\lVert x_l-x_m \right\rVert < \frac{s}{N} \right\} = 2s \end{equation*} for all $s>0$. In this note we show that if the convergence in the above expression is - in a certain sense - fast, then this implies a small discrepancy for the sequence $\{x_n\}_{n \geq 1}$. As an easy consequence it follows that every sequence with Poissonian pair correlations is uniformly distributed in $[0,1)$.
Van der Corput and Halton sequences are well-known low-discrepancy sequences. Almost twenty years ago Ninomiya defined analogues of van der Corput sequences for $β$-numeration and proved that they also form low-discrepancy sequences if $β$ is a Pisot number. Only very recently Robert Tichy and his co-authors succeeded in proving that $\boldsymbolβ$-adic Halton sequences are equidistributed for certain parameters $\boldsymbolβ=(β_1,\ldots,β_s)$ using methods from ergodic theory. In the present paper we continue this research and give discrepancy estimates for $\boldsymbolβ$-adic Halton sequence…
Equidistribution of number theoretic point sets in the torus and the sphere in Wasserstein metric and $L^2$ discrepancy
该研究确定了环面与球面上Kronecker序列、Fibonacci格点等算术点集在Wasserstein度量和L²偏差下的等分布速率，证明了两类度量间的严格不等式及对应最优性关联。
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…
On the one-sided boundedness of the local discrepancy of $\{nα\}$-sequences
本文研究无理数α对应序列{nα}在有理数区间上的局部偏差单侧有界性，推导出若干充要条件，并用拓扑性质刻画满足该性质的α集合大小。
Effektive Abschaetzungen fuer den Gitterrest gewisser ebener und dreidimensionaler Bereiche
某些平面和三维区域格点差异的有效估计。本文给出了三维欧几里得空间中o-对称椭圆盘和椭球的格点差的显式常数估计，以及具有有限非零高斯曲率的光滑边界的三维凸体的格点差的显式常数估计.
Applications of geometric discrepancy in numerical analysis and statistics
本文探讨了几何偏差度量与数值分析、统计学应用之间的联系，包括球面上点分布、接受-拒绝算法及特定马尔可夫链蒙特卡罗算法。
Some Conjectures on the Number of Primes in Certain Intervals
本文对素数作了一些比现有文献中更尖锐的分析。首先，我们描述了我们的研究勒让德的猜想，这仍然是未解决的。接下来，我们表明，布罗卡猜想可以证明假设我们的改进版本的勒让德猜想。最后，我们锐化了素数的Bertrand公设。我们的研究结果得到了广泛的实证研究的支持。
L_p- and S_{p,q}^rB-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases
本文研究了对称van der Corput序列和修改Hammersley点集在任意基底中的局部偏差，给出了Besov空间中的偏差范数上界，从而得到L_p-偏差界，并证明了序列在L_p-和S_{p,q}^rB-偏差上达到最优阶。
Let $α>1$ be an irrational number of finite type $τ$. In this paper, we introduce and study a zeta function $Z_α^\sharp(r,q;s)$ that is closely related to the Lipschitz-Lerch zeta function and is naturally associated with the Beatty sequence ${\mathcal B}(α):=(\lfloorαm\rfloor)_{m\in{\mathbb N}}$. If $r$ is an element of the lattice ${\mathbb Z}+{\mathbb Z}α^{-1}$, then $Z_α^\sharp(r,q;s)$ continues analytically to the half-plane $\{σ>-1/τ\}$ with its only singularity being a simple pole at $s=1$. If $r\not\in{\mathbb Z}+{\mathbb Z}α^{-1}$, then $Z_α^\sharp(r,q;s)$ extends analytically to the …
Random walks with badly approximable numbers
使用差异度量，我们分析了随机游动在d旋转产生的圆上的收敛速率，并建立了尖锐的速度，表明R^d中的d-元组的可近似性差会产生最快收敛的游动。我们使用差异度量，因为行走不会在全变差中收敛。对于可近似性差的d-元组，差异的上下界为（常数）k^（-d/2），其中k是随机游动的步数。我们展示了常数如何依赖于d元组。
On Certain Sums Related to the Largest Odd Divisor
本文考虑与“最大奇因子”有关的有限和，利用简单的思想和递推关系，得到了这类和的精确上界和下界.
The Erdős discrepancy problem, now a theorem by T. Tao, asks whether every sequence with values plus or minus one has unbounded discrepancy along all homogeneous arithmetic progressions. We establish weighted variants of this problem, for weights given either by structured sequences that enjoy some irrationality features, or certain random sequences. As an intermediate result, we establish unboundedness of weighted sums of bounded multiplicative functions and products of shifts of such functions. A key ingredient in our analysis for the structured weights, is a structural result for measure pr…
We prove a sharp decoupling for a certain two dimensional surface in R^9. As an application, we obtain the full range of expected estimates for the cubic Parsell-Vinogradov system in two dimensions.
Discrepancy and numerical integration on metric measure spaces
研究度量测度空间上数值积分的误差，分析单函数和潜在函数类的最坏情况误差，利用Marcinkiewicz-Zygmund不等式和自定义函数空间定义，证明小L^p偏差点分布的存在性。
An important result of H. Weyl states that for every sequence $\left(a_{n}\right)_{n \geq 1}$ of distinct positive integers the sequence of fractional parts of $\left(a_{n} α\right)_{n\geq 1}$ is uniformly distributed modulo one for almost all $α$. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy of $\left(\left\{a_{n} α\right\}\right)_{n \geq 1}$ for almost all $α$. In particular it is very difficult to give sharp lower bounds for the speed of convergence. Until now this was only carried out for lacunary sequences $\left(a_{n}\righ…
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