Asymptotic formula, polynomial, irrational number and uniformly distribution

Problem 1 Given a irrational number α and two polynomials with positive integer coefficients P(n),Q(n), is it possible to get the asymptotic estimate and reasonable error term for: lim In the case when Q(n)=an+b, by Weyl method or van der curput trick it is not difficult to establish such a estimate at least for the … Read more

Algebraic integers whose matrix representations have singular values in an interval

Let $K$ be a finite extension of $\mathbb{Q}$. Let $\mathcal{O}(K)$ be the ring of integers of $K$. Let $\omega_1,\ldots,\omega_n$ be an integral basis for $K$ over $\mathbb{Q}$. For each $a \in K$, multiplication by $a$ is a linear map from $K$ to $K$. Let $M_a$ be matrix for this map with respect to the given … Read more

Hermite Lindemann and transcendental reals

This is about compass and straightedge constructions, although I suspect nothing changes if we expand to real numbers algebraic over the rationals. Anyway, let E be the “constructible numbers,” meaning the smallest subfield of the reals such that, if x>0,x∈E, then √x∈E. So the elements are towers of square roots, add and mix those together. … Read more

More about Roth’s theorem: bound for the constant and multidimensional case

For a real number $x$, we denote $$ \|x\|=\inf_{m\in {\Bbb Z}}|x+m|.$$ Problem 1: Roth’s theorem states that given any irrational algebraic number $\alpha$ and for any $\epsilon>0$, there exists a constant $C(\alpha,\epsilon)$ such that $$\|q\alpha\|>\frac{C(\alpha,\epsilon)}{q^{1+\epsilon}}$$ for each positive integer $q$. Is there an explicit bound for $C(\alpha,\epsilon)$$\ ?$ Problem 2: Let $t=(t_1,\cdots, t_k)$ be a … Read more

Inequality due to Siegel (assumptions) and upper bounds on number field discriminants

In Siegel’s 1969 paper, Abschätzung von Einheiten, on page 73, he states the inequality log√d≤n−1+n2logπ+r2log2(∗) and compares with the bound due to Minkowski that n−112n−log√2πn−r2log(4π)≤log√d where n=[Z:Q] is the degree of a fixed, but arbitrary number field, Z (here Z is not the integers, at least not if I’m reading the German correctly), d is … Read more

Diophantine approximations of ratios of transcendental numbers

I am looking for good diophantine approximations for a specific class of irrational numbers. Let e2πiθ be a complex algebraic number. I would like a result to the effect that θ can be approximated well; more specifically, for any constant k, I would like for the inequality |nθ−m|<1kn to have infinitely many integer solutions in … Read more

Liouville’s Theorem in Diophantine Approximation

Liouville’s Theorem states that for any algebraic $\alpha \in \mathbb{R}$ of degree $n$, there exists a positive constant $c:=c(\alpha)$ such that $$\left\lvert\alpha-\frac{p}{q}\right\rvert>\frac{c}{q^n}$$ for any $p \in \mathbb{Z}$ and $q \in \mathbb{N}.$ One can find an effective lower bound for $c(\alpha).$ In the special case that $\alpha$ is a quadratic irrational, Exercise 27 in the following … Read more

Primes in generalized fibonacci sequences

In C. McMullen’s Uniformly Diophantine numbers in a fixed real quadratic field generalized Fibonacci sequence are defined as follows: $f_0=0,f_1=1,f_m=tf_{m-1}-nf_{m-2}$ where some fixed $t\in \mathbb Z$ and $n$ is $+1$ or $-1$ and $t^2-4n>0$. For example, for $t=1,n=-1$ we get the usual Fibonacci sequence. My question: Does there exist $t,n$ such that the resulting Fibonacci … Read more

On the set of good approximators in the sense of Dirichlet’s theorem

This question came up when thinking about an older question that hasn’t been answered as of now. Let N be the set of positive integers. If α∈R, we say q∈N is good for approximating α if there is p∈Z such that |α−pq|<1q2, and denote the set of those positive integers by G_\alpha. The approximation theorem … Read more

A particular Diophantine approximation of π/2\pi/2

I have asked this question in math.stackexchange without any answer, so I have decided to post it here too. Recently I was playing around with the sequence 1nsin(n), n∈N. After some computations, I was led to the following question: let pn,qn be two sequences of natural numbers such that |pnqn−π2|<1q2n. Can we find a subsequence of … Read more