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### On a sequence of integers

I recall a well-known theorem due to Minkowski. Theorem. If $\theta$ is irrational and $\alpha$ is not of the form $\alpha = m\theta+n$ for some integers $m$ and $n$, then there are infinitely many ...
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### On cluster points of a particular sequence

This is the sequel of a previous question. Let us consider the sequence $$\xi_n = 2n \{n\xi\}-n,$$ where $\xi>0$ is a given real irrational number and $\{\cdot\}$ is the fractional part. Do ...
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### Precise asymptotic of diophantine approximation

I know that every irrational number $\xi$ can be approximated by rational numbers in such a way that $$\left| \xi - \frac{p}{q} \right| \leq \frac{c}{q^2}$$ for infinitely many choices of $p$ and ...
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### On the density of the sequence $\{n \{n \xi \} \}_n$

I have a question that I can't manage to answer myself. It comes from some work in PDE theory, but it is related to analytic number theory. Let us say that we have an irrational number $\xi$. The ...
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### A particular Diophantine approximation of $\pi/2$

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 $$\frac{1}{n\sin(n)},\ n\in\mathbb{N}.$$ ...
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Let $A,B$ be two rational rotations: $$A = \left[\begin{array}{rcc} \frac{3}{5} & \frac{4}{5} & 0 \\ -\frac{4}{5} & \frac{3}{5} & 0 \\ 0 & 0 & 1 \end{array}\right] ... 2answers 1k views ### Approximating any integer by multiples of 2 and 3 Given any integer n sufficiently large, I want to prove (or disprove) that there exists another integer m\ge n with the form m=2^a3^b (a,b are no negative integers) such that m-n=o(n), i.e., ... 0answers 169 views ### Diophantine approximation in the Julia set Let f : \mathbb{CP}^1 \to \mathbb{CP}^1 be a rational map of degree q > 1; or just a quadratic binomial z^2 + c, if one prefers. The Julia set J_f is the closure of the repelling periodic ... 6answers 8k views ### Source and context of \frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)? Possibly the most striking proof of Archimedes's inequality \pi < 22/7 is an integral formula for the difference:$$ \frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2}, $$where the ... 1answer 89 views ### approximation of products of polynomials I am wondering whether the following can be proved: Suppose p(z) and q(z) are polynomials of degree n with real coefficients and leading coefficient 1. Moreover, they have quite different ... 4answers 1k views ### Small quotients of smooth numbers Assume that N=2^k, and let \{n_1, \dots, n_N\} denote the set of square-free positive integers which are generated by the first k primes, sorted in increasing order. Question: what is a good ... 0answers 1k views ### Is there a limit of \cos (n!)? [closed] I encountered a problem today to prove that \cos (n!) does not have a limit. I have no idea how to do it formally. Could someone help? The simpler the proof (by that I mean less complex theorems are ... 0answers 91 views ### Jarník-Besicovitch and outer measure The set A_\tau of irrational numbers x which are \tau-approximable, i.e., that satisfy the estimate$$\left|x - \frac{p}{q}\right| \leq \frac{1}{q^\tau}$$for infinitely many rationals p/q, ... 1answer 217 views ### Central binomials and irrationals I suppose this is une cause perdue, but it would be nice if the following held. Let \theta be an irrational, and let c_m = {{2m}\choose m}, the central binomial. For a real number r, let d(r) ... 1answer 102 views ### Sizes and shapes of Dedekind cuts My geometric intuition has failed to tell me that there are different sizes and shapes of Dedekind cuts. I realized it in the course of writing this answer only by doing algebra. If we define a ... 2answers 372 views ### Dirichlet's approximation only using prime power as denominator I am not sure whether this is a suitable question for MO. We know the classical version of Dirichlet's approximation theorem that if x is a real number and Q>0 there exist p,q\in \mathbb{Z} ... 0answers 188 views ### Attractors of arithmetically small points Consider the "points" to be in \mathbb{G}_m(\mathbb{C}) = \mathbb{C}^{\times}. By a sequence of "arithmetically small points" is meant a sequence of pairwise different algebraic points \beta that, ... 0answers 237 views ### Question on some coverings of the euclidean space Let L be a maximal integral lattice in the euclidean (\mathbf R^{8m},q) (thus the associated bilinear form b(u,v)=q(u+v)-q(u)-q(v), once restricted to L, takes values in 2\mathbf Z and has ... 1answer 184 views ### Can a product of conjugates be a Pisot number again? Let p(X) \in \mathbb{Z}[X] be an irreducible polynomial, and let \alpha_1 \dots, \alpha_n be its roots in \mathbb{C}. Suppose that \alpha_1 is a Pisot number, that is, \alpha_1 \in ... 2answers 549 views ### Does the Galois group of a Pisot polynomial contain the alternating group? Let n \in \mathbb{N}, and let p(X) \in \mathbb{Z}[X] be a monic polynomial of degree n. Suppose that exactly one complex root of p is of modulus > 1, and that the remaining n-1 roots of ... 2answers 2k views ### How often are irrational numbers well-approximated by rationals? Suppose x\in \mathbb{R} is irrational, with irrationality measure \mu=\mu(x); this means that the inequality |x-\frac{p}{q}|< q^{-\lambda} has infinitely many solutions in integers p,q if ... 3answers 371 views ### “Most Similar Vector Problem” on an Integer Lattice? I am currently working on problem that I think could be expressed as an integer lattice problem. Given u \in \mathbb{R}^n and a bounded integer lattice L = \mathbb{Z}^n \cap [-M,M]^n I would like ... 0answers 147 views ### binomial coefficients and irrationals The following, probably either currently impossible to deal with, or having a negative solution, arose from an ergodic theory question, presumably itself currently intractible. I am not a number ... 4answers 6k views ### If 2^x and 3^x are integers, must x be as well? I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let x be a positive real number. If n^x is an integer for ... 1answer 1k views ### Why is there a \sqrt{5} in Hurwitz's Theorem? Hurwitz's theorem is an extension of Minkowski's Theorem and deals with rational approximations to irrational numbers. The theorem states: For every irrational number \alpha, there are infinitely ... 0answers 130 views ### Looking for an effective irrationality measure of \pi Most standard summaries of the literature on irrationality measure simply say, e.g., that$$ \left| \pi - \frac{p}{q}\right| > \frac{1}{q^{7.6063}} $$for all sufficiently large q, without giving ... 1answer 504 views ### A weakening of the Littlewood conjecture For real numbers x, let \|x\| denote the distance from x to the nearest integer. Define a function \ell:\mathbb{R}^2\rightarrow\mathbb{R} by ... 0answers 103 views ### Rate of convergence of an algebraic irrational rotation Let \alpha \in \mathbb{S}^1 be an algebraic number with \mathop{\mathrm{arg}}(\alpha)/\pi irrational. Is it possible for the rotation by \alpha to converge exponentially fast to a \xi \in ... 1answer 138 views ### Positive rational numbers as sum of unit fractions [duplicate] Let U = \{\frac{1}{n}: n\in\mathbb{N}, n>0\} be the set of unit fractions. For integers m,n>0 there is always a finite subset S\subseteq U such that \frac{m}{n} = \sum_{u\in S} u, see ... 1answer 133 views ### Simultaneous approximation by rationals with relatively prime numerators The following seems hard to me (or perhaps just not true), but perhaps I am mistaken. It is known that given irrational numbers x_1 and x_2, there are infinitely many simultaneous rational ... 0answers 56 views ### Big Omega result about number of totally positive integers with fixed trace There is much literature on the study of N_a, the number of totally positive integers with fixed trace a in a totally real field. That number has a natural geometric approximation G_a, and we ... 3answers 648 views ### Is the infimum of Salem numbers > 1? BACKGROUND A Salem number is an algebraic integer \theta such that all the Galois conjugates of \theta are \leq 1 in absolute value, and at least one of them lies on the unit circle. Their ... 2answers 521 views ### For what real t is \{n^t : n \geq 1\} linearly independent over \mathbb{Q}? It's straightforward that t must be irrational. I have googled many variations of this question and browsed through some books on transcendental number theory. There is much that is said about when ... 2answers 1k views ### Solved cubic Thue equation Hi everybody. I need to know if the cubic Thue equation x^3 + x^2y + 3xy^2 - y^3 = \pm 1 is completely solved. I know that there are effective algorithms to solve any cubic Thue equation and that ... 2answers 208 views ### approximate two different real numbers to order \frac{1}{z^{3/2}} I took this result from Minkowski's book on Geometry of numbers: Two arbitrary real quantitites a and b may be made to approach as near as we wish in value the two fractions \frac{x}{z} and ... 0answers 144 views ### Linear forms with best approximation vectors lying in a subspace Setup: For u \in \mathbb{R}^n, let \rho(u) be the Euclidean length, \sqrt{u_1^2 + \ldots + u_n^2}. For x \in \mathbb{R} let \|x\| = \min_{k \in \mathbb{Z}} |x - k|, and for x \in ... 0answers 120 views ### A question related to metric Diophantine approximation In metric Diophantine approximation you are often interested in finding conditions on (\phi(q))_{q \geq 1} which guarantee that$$ \left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q} $$has ... 1answer 131 views ### Khintchine theorem - necessity of monotonicity in divergence condition Hi, I'm trying to get a hold of Khintchine theorem in metric Diophantine approximation. Right now I'm interested in the divergence condition, namely: If \sum_{q=1}^\infty\psi(q) = \infty >  ... 2answers 134 views ### Comparing the Rational Approximability of Infinite Continued Fractions It is known, that \phi := \frac{sqrt(5)-1}{2}, is the number, that is hardest to approximate by rationals (cf e.g. the section properties of the golden ratio \phi here: ... 4answers 3k views ### Massive cancellations Let A=\{a_1,\ldots,a_k\} be a fixed, finite set of reals. Let S_A(n) be the set of all reals that are expressible as the sum of at most 2^n terms, where each term is a product of at most n ... 1answer 404 views ### Is there a bounded sequence of points in the plane with pairwise distances at least 1/\sqrt{|i-j|}? Previously I have mentioned the following problem in an addition to the list of Contest problems with connections to deeper mathematics.. Is there an infinite bounded sequence (P_n) \subset ... 2answers 391 views ### Rate of convergence of an irrational rotation Let \alpha, \beta \in \mathbb{R}. Let \{x\} denote the fractional part of x and let \|x\| = \min(\{x\}, 1-\{x\}). If we assume that \alpha is irrational, then there exists an increasing ... 1answer 248 views ### Computing all “suboptimal” rational approximations to \pi/2 I have an irrational number \alpha (\alpha=\frac\pi2), and I would like to determine all integers n\in[1,N] (N=10^{16}) that satisfy$$ n \epsilon(n)^2 \leq \tau $$where \tau is a known ... 0answers 334 views ### How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices? Title question description: Select two lattices \Lambda_1 and \Lambda_2 (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ... 0answers 361 views ### Is simultaneous diophantine approximation (in a weaker sense) NP hard? The traditional problem of simultaneous diophantine approximation is: Given a set of rational numbers g_1,\ldots,g_d, an integer N, and a rational \gamma>0, is there an integer W with ... 0answers 90 views ### Diophantine approximation in \mathbb{G}_m^r with approximants restricted to a finiteley generated subgroup Faltings, in the same paper (Diophantine approximation on abelian varieties, Ann. Math. 1991) in which he proved his famous big theorem,'' proved also that at any place v of a number field K and ... 1answer 157 views ### Inequality due to Siegel (assumptions) and upper bounds on number field discriminants In Siegel's 1969 paper, Abschätzung von Einheiten, on page 73, he states the inequality$$\log\sqrt d\le n-1+{n\over 2}\log\pi+r_2\log 2\qquad (*)$$and compares with the bound due to Minkowski that ... 0answers 100 views ### Conjectural growth rate for ergodic sums of logarithms Let \theta, \phi \in [0,1), and consider the sums$$ S_n(\theta,\phi)=\sum_{k=0}^n \log|e^{2\pi i (k\theta+\phi)}-1|.  The possible boundedness from above of such sums plays a key role in ...
Let $U=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{-\frac{1}{n}: n\in\mathbb{N}\}$ be the set of positive and negative unit fractions. Are there positive integers $m<n \in \mathbb{N}$, such that for ...
### Lower bound on the irrationality measure of $\pi$
There seems to be a lot of work on the upper bound for the irrationality measure of $\pi$, but I could not find anything on a lower bound except the general $\mu(\pi)\geq2$. Looks like it is the best ...