Related papers: Absolute Bounds for Ergodic Deviations of Linear F…

We study almost sure limiting behavior of extreme and intermediate order statistics arising from strictly stationary sequences. First, we provide sufficient dependence conditions under which these order statistics converges almost surely to…

This paper investigates the optimal ergodic sublinear convergence rate of the relaxed proximal point algorithm for solving monotone variational inequality problems. The exact worst case convergence rate is computed using the performance…

Let $d_N=ND_N(\omega)$ be the discrepancy of the Van der Corput sequence in base $2$. We improve on the known bounds for the number of indices $N$ such that $d_N\leq \log N/100$. Moreover, we show that the summatory function of $d_N$…

For an ergodic hyperbolic measure $\omega$ of a $C^{1+{\alpha}}$ diffeomorphism, there is an $\omega$ full-measured set $\tilde\Lambda$ such that every nonempty, compact and connected subset $V$ of $\mathbb{M}_{inv}(\tilde\Lambda)$…

Let $\mathbf{A}=\{A_i\}_{i=1}^{\infty}$ be a sequence of sets with each $A_i$ being a non-empty collection of $0$-$1$ sequences of length $i$. For $x\in [0,1)$, the maximal run-length function $\ell_n(x,\mathbf{A})$ (with respect to…

For $V\sim \alpha \log\log T$ with $0<\alpha<2$, we prove \[ \frac{1}{T}\text{meas}\{t\in [T,2T]: \log|\zeta(1/2+ {\rm i} t)|>V\}\ll \frac{1}{\sqrt{\log\log T}} e^{-V^2/\log\log T}. \] This improves prior results of Soundararajan and of…

Let $(a_n)_{n \in \mathbb{N}}$ be a Hadamard lacunary sequence. We give upper bounds for the maximal gap of the set of dilates $\{a_n \alpha\}_{n \leq N}$ modulo 1, in terms of $N$. For any lacunary sequence $(a_n)_{n \in \mathbb{N}}$ we…

Answering an informal question of K. Park, we show that by fixing some irrational alpha to have a particular standard continued fraction expansion, we may force the associated discrepancy sequences for all x in [0,1), which track the…

We study the extreme $L_p$ discrepancy of infinite sequences in the $d$-dimensional unit cube, which uses arbitrary sub-intervals of the unit cube as test sets. This is in contrast to the classical star $L_p$ discrepancy, which uses…

Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n x. We prove the following variational inequality in the case where T is power bounded…

An $n$-vertex $k$-uniform hypergraph $G$ is $(d,\alpha)$-degenerate if $m_1(G)\le{d}$ and there exists a constant $\varepsilon >0$ such that for every subset $U\subseteq{V(G)}$ with size $2\le|U|\le{\varepsilon n}$, we have…

We consider a $\varphi$-rigidity property for divergence-free vector fields in the Euclidean $n$-space, where $\varphi(t)$ is a non-negative convex function vanishing only at $t=0$. We show that this property is always satisfied in…

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite-difference equations with random, possibly non-symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic…

Given an irreducible subshift of finite type X, a subshift Y, a factor map \pi : X \to Y, and an ergodic invariant measure \nu on Y, there can exist more than one ergodic measure on X which projects to \nu and has maximal entropy among all…

Let \((a_n)_{n \in \mathbb{N}}\) be a lacunary sequence of integers satisfying the Hadamard gap condition. For any fixed dimension $d \geq 1$, we establish asymptotic upper bounds for the maximal gap in the set of dilates…

In this paper, we establish a noncommutative maximal inequality for ergodic averages with respect to the set $\{k^t|k=1,2,3,...\}$ acting on noncommutative $L_p$ spaces for $p>\frac{\sqrt{5}+1}{2}$.

Given a $\sigma$-finite infinite measure space $(\Omega,\mu)$, it is shown that any Dunford-Schwartz operator $T:\,\mathcal L^1(\Omega)\to\mathcal L^1(\Omega)$ can be uniquely extended to the space $\mathcal L^1(\Omega)+\mathcal…

Let $G = (V,E)$ be a connected directed graph on $n$ vertices. Assign values from the set $\{1,2,\dots,n\}$ to the vertices of $G$ and update the values according to the following rule: uniformly at random choose a vertex and update its…

The \emph{signed series} problem in the $\ell_2$ norm asks, given set of vectors $v_1,\ldots,v_n\in \mathbf{R}^d$ having at most unit $\ell_2$ norm, does there always exist a series $(\varepsilon_i)_{i\in [n]}$ of $\pm 1$ signs such that…

We determine the maximum number $N_\alpha(d)$ of equiangular lines with fixed angle $\arccos\alpha$ for $\alpha = 1/(1+2\sqrt2)$ in $d$-dimensional Euclidean space: $2,3,4,6,8,10,14,15,16,17,18,20,22$ for $d \in \{2,\dots,14\}$, and…