Questions tagged [diophantine-approximation]

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Lower bound for a sequence of numbers

Let $b_1,\cdots,b_s$ be not all zero complex numbers, $a_1,\cdots,a_s$ be distinct numbers. Can one assert there exists a constant $c>0$ such that for any positive integer $n$ $$\left \lvert \sum_{...
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Almost modularity of Belyi curves and etale fundamental group of non-Belyi curves

Belyi's theorem states that every curve defined over $\mathbb{\bar Q}$ is almost modular (obtained from $\mathbb{H}^2/\Gamma,\ \Gamma$ a finite index subgroup of $PSL(2,\mathbb{Z})$), after ...
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Liouville numbers with some "special" convergents

Recall that a real number $\ell$ is called a Liouville number, if there exists an infinite sequence of rational numbers $(p_n/q_n)_n$ for which $$ 0<\left|\ell-\frac{p_n}{q_n}\right|<\dfrac{1}{...
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Does this approximate linear Diophantine Equation have bounded number of solutions?

Consider the linear diophantine equation $$\alpha u+\beta v =r+ \delta$$ where $\alpha,\beta,r\in\mathbb Q$ are known and their binary expansion has $O(k)$ bits to exactly represent them and $\delta\...
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-2 votes
1 answer
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On a criterion for unimodular matrix [closed]

A matrix $$\begin{bmatrix}w &x \\\ y &z\end{bmatrix}\in\mathbb Z^{2\times 2}$$ is unimodular if $$|wz-xy|=1$$ holds. Is there a parametrization of such matrices with $2wy>(wz+xy)$ and $2xz&...
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1 answer
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Almost Diophantine approximation

We have an algebraic number $a$ and a real number $b$. Can the following inequality have infinitely many solutions for $n \in \mathbb{N}$? $$ \{an\} \in [b - \frac{1}{2^n}, b + \frac{1}{2^n}] $$ Here $...
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Dyson's lemma implies index is small (in proving Roth's theorem)

I am reading the proof of Roth's theorem in Hindry-Silverman's book. In there they used Roth's lemma. I think it is well known that the step of Roth's lemma could be replaced by Dyson's lemma to show ...
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Systematic integer approximations of $\ \frac n{\log(n)} $

MO-post             Why Is $\frac{163}{\operatorname{ln}(163)}$ a Near-Integer? presents the topic in a somewhat chaotic manner. However, this topic should admit a regular diophantine analysis of the ...
1 vote
1 answer
134 views

Distribution of $\alpha n^2/q$ modulo $1$?

Let $0 \neq \alpha \in [0,1]$ and $q$ a positive integer. Let $||.||$ denote the distance to the closest integer and define $$ N_i(q) = \sum_{ \substack{ -q/2 \leq n \leq q/2 \\ \frac{i}{q} \leq || \...
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Thue-like equations with low-degree polynomial on the left-hand side

There are known methods for finding integer solutions to Thue equations, that is, equations of the form $f(x,y)=0$, where $f$ is an irreducible homogeneous polynomial of degree $d\geq 3$. Are there ...
6 votes
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Equidistribution modulo 1

We know that the time spent by the sequence $na \mod 1$, $n$ ranging from $1$ up to $x$ and $a$ irrational, at any interval of length $\delta$ is approximately $\delta x$. There are known results when ...
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Simple estimation of difference of powers of 2 and powers of 3?

1. Question How to get from the formulas $$ \left| \frac{\log 2}{\log 3} - \frac{p}{q} \right| < c_a\frac{1}{q^{B_a}} \ \ \ \ \ \ \ \ \ \ \ (1.1)$$ and / or $$ \left| \frac{\log 2}{\log 3} - \frac{...
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Measuring the quality of real approximation

Let $r\in [0,1]\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\mathbb{N}_+$ let $$\alpha_r(n)=\min\{\big|r-\...
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Small scalar products of integer vectors with lattice vectors

Producing small solutions to underdetermined, inhomogenous integer linear systems is a hard problem with wide applications in diophantine approximation. A simpler subproblem is the following. Let $\...
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Markov constant of $\pi$

Given any $\epsilon > 0$, are there infinitely many $(a,b) \in \mathbb{Z}^2$ with $(a,b) = 1$ such that $$\left|\pi - \frac{a}{b}\right| < \frac{\epsilon}{b^2}?$$ According to this document, if ...
1 vote
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A question on multiplicative diophantine approximation

Suppose $0 <\alpha <1$ is an irrational, and $0 < \gamma_1, \gamma_2 < 1$ are real numbers satisfying $\gamma_i \notin \mathbb{Z} \alpha + \mathbb{Z}$ for $i=1,2$. Consider the sequence $(\...
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Stability of successive minima with respect to the metric on the space of lattices

Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the ...
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Uniform distribution mod $1$ vs independence of random variables

Let $a_1, \cdots, a_k \in [0, 1)$ be real numbers such that $1, a_1, \cdots, a_k$ are independent over the rational numbers. By the Weyl equidistribution criterion in $k$-dimensions, we know that the ...
2 votes
2 answers
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The range of each of successive minima for all unimodular lattices

Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the ...
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How far away can we get by multiple rounding and unit change?

This question is inspired by xkcd #2585 (Rounding): Let $u_0,\ldots,u_n$ be positive real numbers (we can assume w.l.o.g. that $u_0=1$) or “units”. Consider the following directed graph: its vertices ...
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2 votes
1 answer
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When does a random trigonometric sum approximate $1$?

I am looking for an upper bound $R=R_{n,\varepsilon}$ such that for given $\varepsilon>0$ and real numbers $\alpha_1, \dotsc, \alpha_n$ in, say, $[1,2]$, there is $x\in [1,R]$ such that $$ \frac 1n\...
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6 votes
1 answer
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Direct proof that the set of badly approximable numbers have full Hausdorff dimension without using Schmidt games

A badly approximable number is an $x$ for which there is a positive constant $c$ such that for all rational $p/q$ we have $$\left|{ x - \frac{p}{q} }\right| > \frac{c}{q^2} \ . $$ The set of badly ...
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Simple Diophantine equation

Are there any solutions in positive integers of $x^3 + 1 = (x - k) y^3$? The closest I can get is $19^3 + 1 = 20 \times 7^3$, but $20\gt 19$ so it just misses! For the related $x^3 - 1 = (x - k) y^3$,...
2 votes
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Baby $abc$ conjecture for $n$-th roots

Is there any progress on a “baby $abc$ conjecture” where you restrict attention to rational approximations of $n$-th roots? Let $r/s$ be a very close approximation to $(t/u)^{1/n}$, so that $$ |u\cdot ...
3 votes
1 answer
224 views

number of integers $n$ with $\|n \alpha \|$ small?

Let $\alpha \in \mathbb{R}$ and $N$ a positive integer. I am interested in the quantity $$ D(\alpha, N) := \# \{ n \in [1, N]: \| n \alpha \| < 1/N \}, $$ $\| x \|$ denotes the distance to the ...
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3 votes
2 answers
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The closure of the orbit of an irrational grid contains the fiber

Let $G=\text{SL}(d,\mathbb R)$ and $\Gamma = \text{SL}(d,\mathbb Z)$. The homogeneous space is identified with the space of unimodular lattices, denoted $X_d$. Let $Y_d$ denote the space of unimodular ...
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Does the equation $a^b+b^c+c^a=d^e$ have solutions in $\mathbb {N}$

Here $a,b,c,d,e$ are distinct and all greater than $1$. This question was formerly posted on Math.Stackexchange, precisely here, but seems to be more general than some other tough number theory ...
2 votes
1 answer
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Distance formula for continued fractions

In the book Neverending fractions from Borwein, van der Poorten, Shallit and Zudilin, there is the so called distance formula (Theorem 2.45, p. 43) stated: $$\alpha_1\alpha_2\cdot...\cdot\alpha_n=\...
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What is known about constructively irrational numbers?

Intuitively, a constructively irrational number is one for which we can effectively separate it from any rational number in terms of the latter's denominator. More formally, a constructively ...
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1 answer
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Almost every $m\times n$ real matrix is Dirichlet approximable

Let $\| \cdot \|$ denote the maximum norm in Euclidean spaces. Consider the set $D_{m,n}$ of $m \times n $ real matrices satisfying that the system of inequalities $$\|Aq-p\|^m < \frac{1}{T}, \|q\|^...
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6 votes
1 answer
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A property of rapid sequences of natural numbers

$\newcommand{\IR}{\mathbb R}$ $\newcommand{\IT}{\mathbb T}$ $\newcommand{\w}{\omega}$ $\newcommand{\e}{\varepsilon}$ Taras Banakh and me proceed a long quest answering a question of ougao at ...
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A pathological (?) function involving powers

This is inspired by a recent math.SE question. Given that mathematicians like to come up with theoretical constructs which do not necessarily always have any practical purpose (but sometimes provide ...
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1 vote
1 answer
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Bounding the fractional parts of the $p^{\text{th}}$ roots of $n,n^2,...,n^{p-1}$

EDIT (August 9, 2021): I would like to ask a more general question. The original question that was fully answered is below the line. For a positive real number $x$, denote the fractional part $x-[x]$ ...
11 votes
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Given $2^n - 1 \mid 3^m - 1$, how large must $m$ be compared to $n$?

Let $m,n$ be natural numbers such that $2^n - 1 \mid 3^m - 1$. By results from Bugeaud-Corvaja-Zannier, say Theorem 3 of this paper , we know that for any constant $C > 0$ we must have $m > Cn$ ...
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The analogue of Liouville's inequality in several variables

Liouville's Theorem in Diophantine approximation asserts that for each irrational algebraic number $\alpha$ of degree $d$, we have $$ |q\alpha - p| \gg_\alpha |q|^{1-d}$$ for all integers $p,q \in \...
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Determine if a 2-variable Diophantine equation has a finite or infinite number of solutions

Do there exist an algorithm, which, given a polynomial $P(x,y)$ with integer coefficients, determines whether Diophantine equation $P(x,y)=0$ has finite or infinite number of integer solutions? Famous ...
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What is the sequence of badly approximable numbers to omit in Hurwitz' second Theorem for Diophantine Approximation to obtain better constants?

The well known result of Hurwitz on Diophantine approximation says that for any irrational $\alpha$ there are infinitely many integer numbers $p$ and $q$ such that $$ |\alpha -\frac{p}{q}|<\frac{1}{...
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A specific Diophantine equation related to the congruent number question

Let $n$ be an odd square free natural number. J.B. Tunnel in his 1983 paper, showed that a number $n$ is congruent, if and only if the number of triples of integers satisfying $2x^2+y^2+8z^2=n$ is ...
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What are some interesting problems in the intersection of Diophantine Approx and Algebraic Geometry?

I am a first year graduate student and I am eager to work on irrationality/transcendental proofs of specific numbers like Euler's constant gamma. Because backgrounds for Elliptic Curves include very ...
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1 vote
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Exponentially well-approximable numbers

Let $\alpha \in [0,1]$ be irrational. I'm interested in the decay rate of $\| n \alpha \|$ as $n \to \infty$, where $\| \cdot \|$ denotes the distance to the nearest integer. For example: Dirichlet'...
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Optimal exponent in Dirichlet’s theorem on diophantine approximation

Let $\vec x = (x_1,x_2,\dots, x_k) \in \mathbb{R}^k$. Dirichlet’s theorem guarantees that for each $N$, there exists $(n_0,n_1,n_2,\dots,n_k) \in \mathbb{Z}^{k+1} \setminus \{\vec 0\}$ with $\max(|n_1|...
3 votes
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A randomised variant of the Littlewood conjecture

A famous conjecture of Littlewood asserts that for each $\alpha, \beta \in \mathbb{R}$ and $\varepsilon > 0$, the set $$ A(\alpha,\beta;\varepsilon) = \{ n \in \mathbb{N} \ : \ \Vert \alpha n \Vert ...
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On the degree of irrationality of two irrational numbers and their rational (in)dependence

Let $x$ and $y$ be some irrational numbers. If the degree of irrationality of $x$ is the same as that of $y$, is it necessarily the case that $x$ and $y$ are rationally dependent ? ADDENDUM: What if $...
1 vote
1 answer
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Bounded, aperiodic irrationals with bounded, aperiodic sum

If $q = [q_0;q_1 \dots]$, say $q_i$ is the $i$-th partial quotient of $q$. My question is the following: Can one construct an explicit example of irrational $r,s > 0$ such that $\{ 1,r,s\}$ is $\...
4 votes
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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 ...
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On the number of asymptotic solutions of the linear Diophantine equation

Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation $$ax+by+cz=n.$$ we have Prove that there exists ...
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1 answer
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Find better than $ 4^n\prod_{k=1}^{n-1}\cos^2(k)\sim e^{o(n)}$

Let $$u_n=\prod_{k=1}^{n-1}\cos^2(k)$$ then $$\frac1n \ln(u_n) = \frac1n\sum_{k=0}^{n-1} \ln(\cos^2(k)) \underset{n\to\infty}\longrightarrow \frac1{2\pi} \int_0^{2\pi} \ln(\cos^2(x))\,{\rm d}x = -\ln(...
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Diophantine approximation on spheres

I am trying to read this letter by Sarnak, where, among other things, he discusses the problem of intrinsic diophantine approximation on $S^3 = \left\{\mathbf{x} \in \mathbb{R}^4 : x_1^2+x_2^2+x_3^2+...
4 votes
1 answer
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The Hausdorff codimension of singular matrices vs. the Hausdorff codimension of points with divergent trajectories

Let $G:=SL(m+n,\mathbb R)$ and $\Gamma :=SL(m+n,\mathbb Z)$ and $X:=G/\Gamma$. (1) Let $M$ denote the set of all $m \times n$ matrices with real entries. A matrix $A \in M$ is called $\textit{singular}...
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algebraic number with explicit base two digits

I am looking for an irrational algebraic number $\alpha \in [0,1[$ whose base two expansion $$ \alpha = \sum_{i=1}^\infty {1 \over 2^{\varphi(i)}}, $$ is easily computable. By this I mean $\varphi : {...
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