## 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

## 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