Related papers: $B'$

If $\alpha_1,\ldots,\alpha_r$ are algebraic numbers such that $$N=\sum_{i=1}^r\alpha_i \ne \sum_{i=1}^r\alpha_i^{-1}$$ for some integer $N$, then a theorem of Beukers and Zagier gives the best possible lower bound on $$\sum_{i=1}^r\log…

Let $m\ge 2$ be an integer, $K$ an algebraic number field and $\alpha\in K\setminus \{0,-1\}$ with sufficiently small absolute value. In this article, we provide a new lower bound for linear form in…

In this paper, we study linear forms \[\lambda = \beta_1\mathrm{e}^{\alpha_1}+\cdots+\beta_m\mathrm{e}^{\alpha_m},\] where $\alpha_i$ and $\beta_i$ are algebraic numbers. An explicit lower bound for the absolute value of $\lambda$ is…

We prove that if $\alpha$ is a non-zero algebraic number of degree $d \geq 2$ which is not a root of unity, then $dh(\alpha)>(1/4) (\log(\log (d))/\log(d))^3.

The purpose of this note is to give a short and elementary proof of the fact, that the absolute logarithmic Weil-height is bounded from below by a positive constant for all totally p-adic numbers which are neither zero nor a root of unity.…

For a non-zero algebraic number $\alpha$ of degree $d$, let $h(\alpha)$ denote its logarithmic Weil height. It is known that when $h(\alpha)$ is small, and $d$ is large, the conjugates of $\alpha$ are clustered near the unit circle and have…

Let $n$ be a positive integer. In this paper we estimate the size of the set of linear forms $b_1\log a_1 + b_2\log a_2+...+b_n\log a_n$, where $|b_i|\leq B_i$ and $1\leq a_i\leq A_i$ are integers, as $A_i,B_i\to \infty$.

"The Baron's omni-sequence", B(n), first defined by Khovanova and Lewis (2011), is a sequence that gives for each n the minimum number of weighings on balance scales that can verify the correct labeling of n identically-looking coins with…

A good range of problems on trees can be described by the following general setting: Given a bilinear map $*:\mathbb R^d\times\mathbb R^d\to\mathbb R^d$ and a vector $s\in\mathbb R^d$, we need to estimate the largest possible absolute value…

In a recent letter, new representations were proposed for the pair of sequences ($\gamma,\delta$), as defined formally by Bailey in his famous lemma. Here we extend and prove this result, providing pairs ($\gamma,\delta$) labelled by the…

Let $\mathbb{I}$ denote an imaginary quadratic field or the field $\mathbb{Q}$ of rational numbers and $\mathbb{Z}_{\mathbb{I}}$ its ring of intergers. We shall prove an explicit Baker type lower bound for $\mathbb{Z}_{\mathbb{I}}$-linear…

We show that if $\lambda_1,\ldots,\lambda_k$ are algebraic numbers, then $$|A+\lambda_1\cdot A+\dots+\lambda_k\cdot A|\geq H(\lambda_1,\ldots,\lambda_k)|A|-o(|A|)$$ for all finite subsets $A$ of $\mathbb{C}$, where…

Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review…

In this paper, we establish lower bounds on Weil height of algebraic integers in terms of the low lying zeros of the Dedekind zeta-function. As a result, we prove Lehmer's conjecture for certain infinite non-Galois extensions conditional on…

In this note, we shall give an improved lower bound for the argument of a power of a given algebraic number which has absolute value one but is not a root of unity.

We obtain general criteria for giving a lower bound on the degree of numbers of the form $\prod_{n=1}^\infty \left(1+\frac{b_n}{\alpha_n}\right)$ or of the form $\prod_{m=1}^\infty \left(1+ \sum_{n=1}^\infty…

We show that if $\mathcal{L}_1$ and $\mathcal{L}_2$ are linear transformations from $\mathbb{Z}^d$ to $\mathbb{Z}^d$ satisfying certain mild conditions, then, for any finite subset $A$ of $\mathbb{Z}^d$, $$|\mathcal{L}_1 A+\mathcal{L}_2…

We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the Lambda-calculus…

Let $F \in \mathbf{Z}[\boldsymbol{x}]$ be a diagonal, non-singular quadratic form in $4$ variables. Let $\lambda(n)$ be the normalised Fourier coefficients of a holomorphic Hecke form of full level. We give an upper bound for the problem of…

We generalize the absolute logarithmic Weil height from elements of the multiplicative group of algebraic numbers modulo torsion, to finitely generated subgoups. The height of a finitely generated subgroup is shown to equal the volume of a…