Scaling limit of Fourier-Walsh coefficients (a framework)

Scaling limits in divisible sandpiles: a Fourier multiplier approach

In this paper we complete the investigation of scaling limits of the odometer in divisible sandpiles on $d$-dimensional tori generalising the works Chiarini et al. (2018), Cipriani et al. (2017, 2018). Relaxing the assumption of independence of the weights of the divisible sandpile, we generate generalised Gaussian fields in the limit by specifying the Fourier multiplier of their covariance kernel. In particular, using a Fourier multiplier approach, we can recover fractional Gaussian fields of the form $(-Δ)^{-(1+s)} W$ for $s>0$ and $W$ a spatial white noise on the $d$-dimensional unit torus.

Mandelbrot 级联的调和分析——在向量值鞅背景下

Harmonic analysis of Mandelbrot cascades -- in the context of vector-valued martingales

本文利用向量值鞅理论解决 MCCM 的 Fourier 维数难题，给出精确公式、Salem 判据、多项式 Rajchman 衰减、β=1/2 二阶相变及限制估计。

Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes

We prove that a random linear code over F_q, with probability arbitrarily close to 1, is list decodable at radius (1-1/q-ε) with list size L=O(1/ε^2) and rate R=Ω_q(ε^2/(log^3(1/ε))). Up to the polylogarithmic factor in (1/ε) and constant factors depending on q, this matches the lower bound L=Ω_q(1/ε^2) for the list size and upper bound R=O_q(ε^2) for the rate. Previously only existence (and not abundance) of such codes was known for the special case q=2 (Guruswami, Håstad, Sudan and Zuckerman, 2002). In order to obtain our result, we employ a relaxed version of the well known Johnson bound on…