Related papers: Limits of Boolean Functions on F_p^n
Suppose that f is a boolean function from F_2^n to {0,1} with spectral norm (that is the sum of the absolute values of its Fourier coefficients) at most M. We show that f may be expressed as +/- 1 combination of at most 2^(2^(O(M^4)))…
A permutation sequence $(\sigma_n)_{n \in \mathbb{N}}$ is said to be convergent if, for every fixed permutation $\tau$, the density of occurrences of $\tau$ in the elements of the sequence converges. We prove that such a convergent sequence…
In this article, we prove a new functional limit theorem for the partial sum sequence $S_{[nt]}=\sum_{i=1}^{[nt]}X_i$ corresponding to a linear sequence of the form $X_i=\sum_{j \in \bZ}c_j \xi_{i-j}$ with i.i.d. innovations $(\xi_i)_{i \in…
We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs…
The probabilistic degree of a Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is defined to be the smallest $d$ such that there is a random polynomial $\mathbf{P}$ of degree at most $d$ that agrees with $f$ at each point with high…
A function defined on the Boolean hypercube is $k$-Fourier-sparse if it has at most $k$ nonzero Fourier coefficients. For a function $f: \mathbb{F}_2^n \rightarrow \mathbb{R}$ and parameters $k$ and $d$, we prove a strong upper bound on the…
A sequence of $k$-uniform hypergraphs $H_1, H_2, \dots$ is convergent if the sequence of homomorphism densities $t(F, H_1), t(F, H_2), \dots$ converges for every $k$-uniform hypergraph $F$. For graphs, Lov\'asz and Szegedy showed that every…
Optimal sampling of non band-limited functions is an issue of great importance that has attracted considerable attention. We propose to tackle this problem through the use of a frequency warping: First, by a nonlinear shrinking of…
In this article, we quantify the functional convergence of the rescaled random walk with heavy tails to a stable process.This generalizes the Generalized Central Limit Theorem for stable random variables infinite dimension. We show that…
An equivalent definition of the Fibonacci numbers is that they are the unique sequence such that every integer can be written uniquely as a sum of non-adjacent terms. We can view this as we have bins of length 1, we can take at most one…
Linear processes are defined as a discrete-time convolution between a kernel and an infinite sequence of i.i.d. random variables. We modify this convolution by introducing decimation, that is, by stretching time accordingly. We then…
In an earlier paper the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency…
We establish a pointwise limit theorem for a broad class of pa\-ra\-me\-ter-\-de\-pen\-dent BMO-type seminorms as the parameter tends to zero. By introducing novel BMO-type seminorms, we provide a unified framework that extends several…
The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences $(G_n)$ of graphs in terms of a limiting object which may be represented by a symmetric function $W$ on…
In earlier papers we have developed an algebraic theory of discrete tomography. In those papers the structure of the functions $f: A \to \{0,1\}$ and $f: A \to \mathbb{Z}$ having given line sums in certain directions have been analyzed.…
Action convergence provides a limit theory for linear bounded operators $A_n:L^{\infty}(\Omega_n)\longrightarrow L^1(\Omega_n)$ where $\Omega_n$ are potentially different probability spaces. This notion of convergence emerged in graph…
We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying L\'{e}vy measures. The limit process is a new class of symmetric stable…
We give a general strategy to construct superoscillating/growing functions using an orthogonal polynomial expansion of a bandlimited function. The degree of superoscillation/growth is controlled by an anomalous expectation value of a…
We study the space of functions computed by random-layered machines, including deep neural networks and Boolean circuits. Investigating the distribution of Boolean functions computed on the recurrent and layer-dependent architectures, we…
In this note we study the error term R_{n,L}(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary…