Related papers: Bohr's phenomenon for functions on the Boolean cub…

We give tight bounds on the degree $\ell$ homogenous parts $f_\ell$ of a bounded function $f$ on the cube. We show that if $f: \{\pm 1\}^n \rightarrow [-1,1]$ has degree $d$, then $\| f_\ell \|_\infty$ is bounded by $d^\ell/\ell!$, and $\|…

Let $f(n)$ denote a multiplicative function with range $\{-1,0,1\}$, and let $F(x) = \sum_{n\leq x} f(n)$. Then $F(x)/\sqrt{x} = a\sqrt{x} + b + E(x)$, where $a$ and $b$ are constants and $E(x)$ is an error term that either tends to $0$ in…

In this paper, we show that, for any $\beta \in [1,2]$, a given strictly positive real-valued continuous function on $[0,1]$ whose graph has upper box-counting dimension less than or equal to $\beta $ can be decomposed as a product of two…

In this paper, we investigates the Bohr phenomenon for holomorphic mappings $F$ from the unit ball $\mathbb{B}_X$ of a complex Banach space $X$ into the closure of the unit polydisc $\mathbb{D}^m$ within the space $\mathbb{C}^m$. First, we…

Let $(r_k)$ be an increasing sequence and $(w_k)$ a positive sequence. We study the following question: is it true that for every sequence $(a_k)$ satisfying $\sum_{k=0}^\infty |a_k|^2 w_k^2 < \infty$ there exists a function $f\in…

The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\sum_{n=0}^\infty |a_n|r^n \leq 1$ holds whenever $|\sum_{n=0}^\infty a_nz^n|\leq 1$ in the unit disk $\mathbb{D}$…

It has previously been an open problem whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This problem has been considered within several different…

Let $G$ be a finite abelian group acting faithfully on a finite set $X$. As a natural generalization of the perfect nonlinearity of Boolean functions, the $G$-bentness and $G$-perfect nonlinearity of functions on $X$ are studied by Poinsot…

A symmetric pseudo-Boolean function is a map from Boolean tuples to real numbers which is invariant under input variable interchange. We prove that any such function can be equivalently expressed as a power series or factorized. The kernel…

A Boolean function $f({\vec x})$ is sensitive to bit $x_i$ if there is at least one input vector $\vec x$ and one bit $x_i$ in $\vec x$, such that changing $x_i$ changes $f$. A function has sensitivity $s$ if among all input vectors, the…

Let $f$ be a full-level cusp form for $GL_m(\mathbb Z)$ with Fourier coefficients $A_f(n_1,...,n_{m-1})$. In this paper an asymptotic expansion of Voronoi's summation formula for $f$ is established. As applications of this formula, a…

We prove that the Bohr' radius for large functions is $e^{-\pi }.$

Let $P_{2k}$ be a homogeneous polynomial of degree $2k$ and assume that there exist $C>0$, $D>0$ and $\alpha \ge 0$ such that \begin{equation*} \left\langle P_{2k}f_{m},f_{m}\right\rangle_{L^2(\mathbb{S}^{d-1})}\geq \frac{1}{C\left(…

The primary objective of this paper is to establish several sharp versions of improved Bohr inequality, refined Bohr-type inequality, and refined Bohr-Rogosinski inequality for the class of $K$-quasiconformal sense-preserving harmonic…

A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = A+A = \{a + b \ | \ a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. We prove that the number of sumsets in $\mathbb{F}_2^n$ is asymptotically…

Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function $f$, $\bullet \quad \mathrm{deg}(f) = O(\widetilde{\mathrm{deg}}(f)^2)$: The degree of $f$ is at most quadratic in the approximate degree of $f$.…

Consider a subset $A$ of $\mathbb{F}_p^n$ and a decomposition of its indicator function as the sum of two bounded functions $1_A=f_1+f_2$. For every family of linear forms, we find the smallest degree of uniformity $k$ such that assuming…

The Fourier-Entropy Influence (FEI) Conjecture states that for any Boolean function $f:\{+1,-1\}^n \to \{+1,-1\}$, the Fourier entropy of $f$ is at most its influence up to a universal constant factor. While the FEI conjecture has been…

If a function $f:\mathbb{R}\to\mathbb{R}$ can be represented as the sum of $n$ periodic functions as $f=f_1+\dots+f_n$ with $f(x+\alpha_j)=f(x)$ ($j=1,\dots,n$), then it also satisfies a corresponding $n$-order difference equation…

It is well known that the Fourier--Bohr coefficients of regular model sets exist and are uniformly converging, volume-averaged exponential sums. Several proofs for this statement are known, all of which use fairly abstract machinery. For…