Related papers: On infinite multiplicative Sidon sets

We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power…

We prove, in particular, that for any finite set of real numbers A with |A/A| \ll |A| one has |A-A| > |A|^{5/3 - o(1)}. Also we show that |3A| > |A|^{2-o(1)} in the case.

Let $\pi$ be a polarized, regular algebraic, cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb{A}_F)$ where $F$ is totally real or imaginary CM, and let $(\rho_\lambda)_\lambda$ be its associated compatible system of…

In this paper, we study the explicit geometry of threefolds, in particular, Fano varieties. We find an explicitly computable positive integer $N$, such that all but a bounded family of Fano threefolds have $N$-complements. This result has…

To each complex number $\lambda$ is associated a representation $\pi_\lambda$ of the conformal group $SO_0(1,n)$ on $\mathcal C^\infty(S^{n-1})$ (spherical principal series). For three values $\lambda_1,\lambda_2,\lambda_3$, we construct a…

In this paper we show two results. In the first result we consider $\lambda_n-n=\frac{A}{n^\alpha}$ for $n\in\mathbb N$; if $\alpha>1/2$ and $0<A<\frac{1}{\pi\sqrt{2 \sqrt{2}\zeta(2\alpha)}}$, the system $\left\{\operatorname{sinc}(…

Let $\Psi :[1,\infty )\rightarrow \mathbb{R}_{+}$ be a non-decreasing function, $a_{n}(x)$ the $n$'{th} partial quotient of $x$ and $q_{n}(x)$ the denominator of the $n$'{th} convergent. The set of $\Psi $-Dirichlet non-improvable numbers…

We give an elementary proof that the sum of the digits of $2^n$ in base 10 is greater than $\log_4 n$. In particular, the limit of the sum of digits of $2^n$ is infinite.

In this paper we count the number $N_3^{\text{tor}}(X)$ of $3$-dimensional algebraic tori over $\mathbb{Q}$ whose Artin conductor is bounded by $X$. We prove that $N_3^{\text{tor}}(X) \ll_{\varepsilon} X^{1 + \frac{\log 2 +…

Motivated by questions in number theory, Myerson asked how small the sum of 5 complex nth roots of unity can be. We obtain a uniform bound of O(n^{-4/3}) by perturbing the vertices of a regular pentagon, improving to O(n^{-7/3}) infinitely…

Let $\mathcal{A}$ be a finite set of $d\times d$ matrices with integer entries and let $m_n(\mathcal{A})$ be the maximum norm of a product of $n$ elements of $\mathcal{A}$. In this paper, we classify gaps in the growth of…

The process of alternately row scaling and column scaling a positive $n \times n$ matrix $A$ converges to a doubly stochastic positive $n \times n$ matrix $S(A)$, called the \emph{Sinkhorn limit} of $A$. Exact formulae for the Sinkhorn…

For each natural number $d$, we introduce the concept of a $d$-cap in $\mathbb{F}_3^n$. A subset of $\mathbb{F}_3^n$ is called a $d$-cap if, for each $k = 1, 2, \dots, d$, no $k+2$ of the points lie on a $k$-dimensional flat. This…

We give several new applications of our theorem on the existence of multiplicity of graded families of ideals as a limit, including a very general Minkowski type inequality for graded families of ideals, a very general formula for existence…

Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, by F\"{u}rer, shows that two $n$-bit numbers can be multiplied via a boolean circuit of size $O(n \lg…

Many identities written by $P=S=C$ are obtained, where $P$ infinite products, $S$ infinite series, and $C$ continued fractions. Such equality is called {\it triplicity}, and it can be used to compute the values of infinite series. It is…

For a permutation $\pi$, let $S_{n}(\pi)$ be the number of permutations on $n$ letters avoiding $\pi$. Marcus and Tardos proved the celebrated Stanley-Wilf conjecture that $L(\pi)= \lim_{n \to \infty} S_n(\pi)^{1/n}$ exists and is finite.…

Re presenting the traditional proof of Srinivasa Ramanujan's own favorite series for the reciprocal of $\pi$ :\begin{equation}\frac{1}{\pi} = \frac{\sqrt{8}}{9801} \sum_{n=0}^{+\infty} \frac{(4n)!}{(n!)^4} \frac{1103 + 26390n}{396^{4n}} \;…

We show that if $\pi$ is the fundamental group of a 4-dimensional infrasolvmanifold then $-2\leq{def(\pi)}\leq0$, and give examples realizing each of these values. We also determine the abstract commensurators of such groups. Finally we…

We prove that for all $n$, simultaneously, we can choose prime filtrations of $R/I^n$ such that the set of primes appearing in these filtrations is finite.