Related papers: Boolean Functions with Small Approximate Spectral …

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)))…

The spectral norm of a Boolean function $f:\{0,1\}^n \to \{-1,1\}$ is the sum of the absolute values of its Fourier coefficients. This quantity provides useful upper and lower bounds on the complexity of a function in areas such as learning…

In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of f is $\|\hat{f}\|_1=\sum_{\alpha}|\hat{f}(\alpha)|$). Specifically, we prove the following results for functions $f:\{0,1\}^n \to…

Using the spectral theorem we compute the Quantum Fourier Transform (or Vacuum Characteristic Function) $\langle \Phi, e^{itH}\Phi\rangle$ of an observable $H$ defined as a self-adjoint sum of the generators of a finite-dimensional Lie…

Suppose that A is a subset of F_2^n of density as close to 1/3 as possible. We show that the A(F_2^n)-norm (that is the sum of the absolute values of the Fourier transform) of the characterstic function of A is bounded below by an absolute…

It is a well-known fact that Riemann Hypothesis will follows if the function identically equal to -1 can be arbitrarily approximated in the norm $\norma{.}$ of $L^{2}([0,1],dx)$ by functions of the form $f(x)=\sum_{k=1}^{n}a_{k}…

Let $f$ be a real-valued, degree-$d$ Boolean function defined on the $n$-dimensional Boolean cube $\{\pm 1\}^{n}$, and $f(x) = \sum_{S \subset \{1,\ldots,d\}} \widehat{f}(S) \prod_{k \in S} x_k$ its Fourier-Walsh expansion. The main result…

A classical result of Rothschild and van Lint asserts that if every non-zero Fourier coefficient of a Boolean function $f$ over $\mathbb{F}_2^{n}$ has the same absolute value, namely $|\hat{f}(\alpha)|=1/2^k$ for every $\alpha$ in the…

We show that if $f$ is an integer-valued function with spectral norm at most $M$ then there are subspaces $V_1,\dots,V_L$ and signs $\sigma_1,\dots,\sigma_L \in \{-1,1\}$ such that $f=\sigma_1 1_{V_1} + \dots + \sigma_L 1_{V_L}$ where $L <…

In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f^2 = I. We describe several generalisations of well-known results in the theory of boolean functions, including…

Boolean functions on the space $F_{2}^m$ are not only important in the theory of error-correcting codes, but also in cryptography, where they occur in private key systems. In these two cases, the nonlinearity of these function is a main…

We show that if $G$ is a finite Abelian group and $f$ is an integer-valued map on $G$ with algebra norm at most $M$ then there is some $L < \exp(M^{4+o(1)})$, cosets of (possibly different) subgroups $W_1,...,W_L$, and $s_1,...,s_L \in…

Let $G$ be a finite abelian group and $A$ a subset of $G$. The spectrum of $A$ is the set of its large Fourier coefficients. Known combinatorial results on the structure of spectrum, such as Chang's theorem, become trivial in the regime…

A function $f:\ \{-1,1\}^n\rightarrow \mathbb{R}$ is called pseudo-Boolean. It is well-known that each pseudo-Boolean function $f$ can be written as $f(x)=\sum_{I\in {\cal F}}\hat{f}(I)\chi_I(x),$ where ${\cal F}\subseteq \{I:\ I\subseteq…

The theorem states that: Every Boolean function can be $\epsilon -approximated$ by a Disjunctive Normal Form (DNF) of size $O_{\epsilon}(2^{n}/\log{n})$. This paper will demonstrate this theorem in detail by showing how this theorem is…

We study Boolean functions with sparse Fourier coefficients or small spectral norm, and show their applications to the Log-rank Conjecture for XOR functions f(x\oplus y) --- a fairly large class of functions including well studied ones such…

The $U^2$ norm gives a useful measure of quasirandomness for real- or complex-valued functions defined on finite (or, more generally, locally compact) groups. A simple Fourier-analytic argument yields an inverse theorem, which shows that a…

Bohr proved that a uniformly almost periodic function $f$ has a bounded spectrum if and only if it extends to an entire function $F$ of exponential type $\tau(F) < \infty$. If $f \geq 0$ then a result of Krein implies that $f$ admits a…

In this note we consider Boolean functions defined on the discrete cube equipped with a biased product probability measure. We prove that if the spectrum of such a function is concentrated on the first two Fourier levels, then the function…

A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = \{a + b : a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. Sumsets are central objects of study in additive combinatorics, featuring in several influential…