Related papers: Reducing the Sarnak Conjecture to Toeplitz systems

We prove that for every integer $k$, there exists $\varepsilon > 0$ such that for every n-vertex graph $G$ with no pivot-minor isomorphic to $C_k$, there exist disjoint sets $A,B \subseteq V(G)$ such that $|A|,|B| \geq \varepsilon n$, and…

Building on the recent work of Johnson (2007) and Yu (2008), we prove that entropy is a concave function with respect to the thinning operation T_a. That is, if X and Y are independent random variables on Z_+ with ultra-log-concave…

J.J. Schaeffer proved that for $any$ induced matrix norm and $any$ invertible $T=T(n)$ the inequality \[\left|\det T\right|\left\Vert T^{-1}\right\Vert \leq\mathcal{S}\left\Vert T\right\Vert ^{n-1}\] holds with…

The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…

The purpose of this work is to bound sofic topological entropy of Toeplitz systems over residually finite groups and to prove the Krieger Theorem about attaining arbitrary entropy by the Toeplitz systems. To achieve these results, we…

We present finitary formulations of two well known results concerning infinite series, namely Abel's theorem, which establishes that if a series converges to some limit then its Abel sum converges to the same limit, and Tauber's theorem,…

Let $\mathcal{S} = \{ \tau_n \}_{n=1}^\infty \subset (0,T)$ be an arbitrary countable (dense) set. We show that for any given initial density and momentum, the compressible Euler system admits (infinitely many) admissible weak solutions…

In this paper, we study the linear mapping which sends the sequence $x=(x_n)_{n \in \mathbb{N}}$ to $y=(y_n)_{n \in \mathbb{N}}$ where $y_n = \sum_{k=1}^\infty f(n/k)x_k$ for $f: \mathbb{Q}^+ \to \mathbb{C}$. This operator is the…

New upper and lower bounds for the $\ell_p (1<p<\infty)$ norms of Cauchy-Toeplitz matrices in the form $T_n=[2/(1+2(i-j))]_{i,j=1}^n$ are derived. Moreover, we give a complete answer to a conjecture proposed by D. Bozkurt.

We study properties of the Weyl pseudometric associated with an action of a countable amenable group on a compact metric space. We prove that the topological entropy and the number of minimal subsets of the closure of an orbit are both…

In this paper analyzes \textit{The Erd\H{o}s-Straus conjecture} asserts that $f$$(n)$ $>$ 0 for every $n$ $\geq$ 2, where $f(n)$ indicates the number of solutions to the Diophantine Equation…

We solve a case of the Abelian Exponential-Algebraic Closedness Conjecture, a conjecture due to Bays and Kirby, building on work of Zilber, which predicts sufficient conditions for systems of equations involving algebraic operations and the…

This is the last paper of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number~$k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at…

A family of disjoint pairs of finite sets $\mathcal{P}=\{(A_i,B_i)\mid i\in[m]\}$ is called a Bollob\'as system if $A_i\cap B_j\neq\emptyset$ for every $i\neq j$, and a skew Bollob\'as system if $A_i\cap B_j\neq\emptyset$ for every $i<j$.…

We prove sufficient conditions for the boundedness and compactness of Toeplitz operators $T_a$ in weighted sup-normed Banach spaces $H_v^\infty$ of holomorphic functions defined on the open unit disc $\mathbb{D}$ of the complex plane; both…

The famous Erdos-Heilbronn conjecture plays an important role in the development of additive combinatorics. In 2007 Z. W. Sun made the following further conjecture (which is the linear extension of the Erdos-Heilbronn conjecture): For any…

In this paper a necessary and sufficient condition is deduced for the close-to-convexity of a cross product of Bessel and modified Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about…

Consider tossing a collection of coins, each fair or biased towards heads, and take the distribution of the total number of heads that result. It is natural to conjecture that this distribution should be 'more random' when each coin is…

We show that the M\"{o}bius function is orthogonal to the Thue-Morse sequence $t(n)$ taken along the Piatetski-Shapiro numbers $\lfloor n^c \rfloor$ for any $1 < c < 2$. Previously this property was established for the subsequence along the…

At present there exist numerous different approaches to results on Toeplitz determinants of the type of Szeg\"o's strong limit theorem. The intention of this paper is to show that Jacobi's theorem on the minors of the inverse matrix remains…