Related papers: An explicit Baker type lower bound of exponential …

Let $\alpha$ be a real number such that $1< \alpha <2$ and let $x_0=x_0(\alpha)$ be a {\rm(}unique{\rm)} positive solution of the equation $$ x^{\alpha-1} -\frac{\pi}{e^2\sqrt{3}}x +1=0. $$ Then we prove that for each positive integer…

Given an isotropic quadratic form over a number field which assumes a value $t$, we investigate the distribution of points at which this value is assumed. Building on the previous work about the distribution of small-height zeros of…

In this paper we study the problem of counting Salem numbers of fixed degree. Given a set of disjoint intervals $I_1,\ldots, I_{k}\subset \left[0;\pi\right]$, $1\leq k\leq m$ let $Sal_{m,k}(Q,I_1,\ldots,I_{k})$ denote the set of ordered…

In the 1990's exponential-type error bounds appeared in the theory of radial basis functions. This kind of error bounds is very powerful. However it only measures the difference between the approximant and approximand. Mathematicians and…

In this paper we compute Gr\"obner bases for determinantal ideals of the form $I_{1}(XY)$, where $X$ and $Y$ are both matrices whose entries are indeterminates over a field $K$. We use the Gr\"obner basis structure to determine Betti…

Let $\mathcal M=\langle K;O\rangle$ be a real closed valued field and let $k$ be its residue field. We prove that every interpretable field in $\mathcal M$ is definably isomorphic to either $K$, $K(\sqrt{-1})$, $k$, or $k(\sqrt{-1})$. The…

We study a linear form in the values of Euler's series $F(t)=\sum_{n=0}^\infty n!t^n$ at algebraic integer points $\alpha_1, \ldots, \alpha_m \in \mathbb{Z}_{\mathbb{K}}$ belonging to a number field $\mathbb{K}$. Let $v|p$ be a…

Classical (or ``global'') Bernstein theory establishes sharp control on entire functions of exponential type that are bounded and real-valued on the real axis. We localize some of this theory to rectangular regions $\{ x+iy: x \in I, 0 \leq…

We prove a lower bound to quantum Max Cut of a graph in terms of the Lov\'asz theta function of its complement. For a graph with $m$ edges, $\text{qmc}(G) \geq \tfrac{m}{4}\big( 1 + \tfrac{8}{3\pi}\tfrac{1}{\vartheta(\bar{G}) -1} \big)$,…

Let $\alpha$ be a complex number. We show that there is a finite subset $F$ of the ring of the rational integers $\mathbb{Z}$, such that $F\left[ \alpha\right] =\mathbb{Z}\left[ \alpha\right]$, if and only if $\alpha$ is an algebraic number…

Let \({\mathbb K}\) be any field, let \(X\subset {\mathbb P}^{k-1}\) be a set of \(n\) distinct \({\mathbb K}\)-rational points, and let \(a\geq 1\) be an integer. In this paper we find lower bounds for the minimum distance \(d(X)_a\) of…

This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of…

In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are `close' (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a…

Bourgain, Konyagin and Shparlinski obtained a lower bound for the size of the product set AB when A and B are sets of positive rational numbers with numerator and denominator less or equal than Q. We extend and slightly improve that lower…

Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly…

We prove over fields of power series the analogues of several Diophantine approximation results obtained over the field of real numbers. In particular we establish the power series analogue of Kronecker's theorem for matrices, together with…

We give lower bounds on the number of effective divisors of degree $\leq g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds and…

Recently the authors showed that the algebraic integers of the form $-m+\zeta_k$ are bases of a canonical number system of $\mathbb{Z}[\zeta_k]$ provided $m\geq \phi(k)+1$, where $\zeta_k$ denotes a $k$-th primitive root of unity and $\phi$…

Let K be a field of characteristic 0 and let n be a natural number. Let Gamma be a subgroup of the multiplicative group $(K^\ast)^n$ of finite rank r. Given $A_2,...,a_n\in K^\ast$ write $A(a_1,...,a_n,\Gamma)$ for the number of solutions…

We derive a lower bound for the subword complexity of the base-$b$ expansion ($b\geq 2$) of all real numbers whose irrationality exponent is equal to 2. This provides a generalization of a theorem due to Ferenczi and Mauduit. As a…