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Questions tagged [diophantine-approximation]

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Recovering integrality by solving rational polynomial systems that approximate real polynomial systems

Given $n$ polynomials $g_1(x_1,\dots,x_{n}),\dots,g_{n}(x_1,\dots,x_{n})\in\mathbb R[x_1,\dots,x_{n}]$ where each of $g_1(x_1,\dots,x_{n}),\dots,g_{n}(x_1,\dots,x_{n})$ is homogeneous of degree $d$ ...
7
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1answer
143 views

Limit of quotients of elements of special Fibonacci matrices

Let $F_n$ be the $n$-th Fibonacci number, started with $F_0=0,F_1=1$, and consider the matrices $$M_n=\pmatrix{F_{n+3} & F_{n+1} \\ F_{n+2} & F_{n}}.$$ Let $$\pmatrix{\alpha_n & \beta_n \\...
7
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1answer
171 views

Distribution of $\{cn^a\}$

Assume that $1<a<2$ and $c\ne 0$ is a real number. What is known about the distribution of the sequence $cn^a$ modulo 1? Say, is it true that for certain $\theta<1$ (depending on $a$ and $c$) ...
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Independence of number fields generated by roots of Littlewood polynomials

Let $\mathcal{R}_d \subset \bar{\mathbb{Q}}$ be the set of all roots of degree $d$ polynomials with $\{-1,0,1\}$ coefficients and $$ c(d) := \min_{\substack{ \alpha, \beta \in \mathcal{R}_d \\ \alpha^...
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0answers
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The Hausdorff dimensions of variations of Jarnik sets

For $\alpha, \beta>3,$ define $$\{(x,y)\in[0,1]\times [0,1]: \|qx\|\le q^{1-\alpha}, \|qy\|\le q^{1-\beta} \quad \text{for infinitely many $ q\in \mathbb{N}$}\}.$$ This set can be regarded as a two ...
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0answers
63 views

Approximation of an irrational point from a given direction

Taking norms to be maximal norm, then the simultaneous version of the Dirichlet's approximation theorem states that given real numbers ${\displaystyle \alpha _{1},\ldots ,\alpha _{d}} $,there are ...
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182 views

average value of j-invariant at infinity

Let $\xi\in\mathbb{R}$ and consider the average value (with respect to hyperbolic length) of the $j$-invariant ($j(z)=q^{-1}+744+196884q+\ldots$, $q=e^{2\pi iz}$) along a geodesic aimed at $\xi$: $$ \...
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0answers
40 views

Discrepancy related independent vector from tensor product?

Here discrepancy is from $(2.4)$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf given by 'The discrepancy $D_N(P) = D_N(x_l,\dots,X_N)$ of the point set $P$ of $N$ points in $\mathbb Z^...
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0answers
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Does a weighted sequence derived from a sequence matching the Siegel lemma BV bound behave as an uniformly random sequence?

Pick integers $r_1',\dots,r_t'$ that achieve the Bombieri Vaaler bound for the Siegel lemma and find an $m$ (it exists by Dirichlet Pigeonhole) such that given prime $T$ gets $m(r_1',\dots,r_t')\equiv(...
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Are there partially algebraic Hecke characters?

$\newcommand{\C}{\mathbb{C}} \newcommand{\U}{\mathbb{U}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}}$ Let $F$ be a number field. Let $\chi\colon \mathbb{A}_F^\...
2
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0answers
115 views

Combination of irrationals

Fix a very small $\epsilon>0$; and irrationals $a_1,a_2>0$. Now suppose we look at all integer combinations of these irrationals which has a small norm; that is, $$ S=\{(m_1,m_2)\in\mathbb{Z}\...
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0answers
124 views

An elementary size bound in number theory?

Given integer $a$ of size $R^\alpha$ with $\alpha\in(0,1)$ and $t$ large primes $R_i$ of roughly same size $R$ what is the smallest $T$ one needs so that there is an integer $0<K<R$ with as ...
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1answer
89 views

Simultaneous rational approximation to transcendental and algebraically independent numbers

I’m interested in the problem of simultaneous rational approximation to $k\geq 2$ numbers $\alpha_1,…,\alpha_k$ in the generic case where the $\alpha_j$ are transcendental and algebraically ...
2
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1answer
111 views

Variations of Dirichlet's theorem on Diophantine approximation

Dirichlet's theorem on Diophantine approximation: For any real number $x$, for integer $N>0$, there exists integers $a$ and $b>0$ with $(a,b)=1$ such that $b\leq N$ and $$|x-\frac a b|<\frac{...
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0answers
178 views

$\{x^n\}$ dense in $[0,1]$ for rational noninteger $x > 1$ [closed]

Is the sequence $\{\{x^0\}, \{x^1\}, \{x^2\}, \cdots \}_{n \geq 0}$ dense in $[0,1]$ for rational noninteger $x > 1$, where $\{x\}$ denotes the fractional part of $x$, i.e $\{ x \} = x - \lfloor x \...
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0answers
104 views

Quantitative Khintchin's theorem

The quantitative version of Kchintchin's theorem proved by W.Schmidt states that for a.e. $x\in[0,1]$ and every positive integer $K$, if we denote the number of solutions $(p,q)$ to the inequality $$|...
4
votes
1answer
225 views

Explicit numbers with square root cancellation in Weyl's exponential sum

I'm interested in examples of real numbers $\alpha$ where we have $$\left| \sum_{n=1}^N \mathrm e(\alpha n) \right| \ll N^{1/2} $$ or perhaps with the weaker estimate with the right side replaced ...
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1answer
149 views

Best possibe bound for the number of solutions of diophantine approximation?

I am interested in an explicit - best possible - bound for the number of solutions of the simple diophantine equation $$\tag{1}|\mu_1 n - \mu_2 m| <1,$$ where $m,n \in \mathbb{Z}$ with $|n| \leq |\...
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163 views

Transcendental Continued Fractions

Liouville famously showed the existence of a transcendental number by considering $\alpha =\sum\limits_{n=0}^\infty10^{-n!}$ and showing that it did not satisfy 'Liouville's criterion' $|\alpha-p/q|\...
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3answers
423 views

Lower bound for the fractional part of $(4/3)^n$

My son, who is 16, is doing some independent research. A lower bound depending on $n$ for $\left\{ \left( \frac{4}{3} \right)^n \right\}=\left( \frac{4}{3} \right)^n-\left\lfloor \left(\frac{4}{3} \...
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2answers
720 views

Rational approximations of $\sqrt{2}$ in $\mathbb{R} \times \mathbb{Q}_7$

Note: this question was updated (2) after GNiklasch's answer was posted, and taking Gro-Tsen's comment into account. The initial question (1) dealt with $\mathbb{Q}_3$. Original post (1). Let's try ...
1
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1answer
110 views

Does the set of Diophantine $m$-tuples has full measure?

We say that an $m$-tuple $\omega=(\omega_1,\ldots,\omega_m)$ satisfies the Diophantine condition of order $\nu \geq 0$ if there is a constant $C>0$ such that for all natural $q$ and integer $p_1,\...
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0answers
41 views

On existence of certain vectors from moduli of structured vectors?

Fix a small ${\epsilon}>0$ and take a very large $n>0$. Pick uniformly random integers $A,B,C,D,A',B',C',D'$ with $A,B$ is coprime, $C,D$ is coprime, $A',B'$ is coprime and $C',D'$ is coprime ...
1
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1answer
107 views

Simultaneous Diophantine Condition and Growth Rate of Convergents Denominators

Let $\omega=(\omega_1,\ldots,\omega_{m})$ be an $m$-tuple of real numbers. Let $|\omega|_{m}:=\sup\limits_{1 \leq j \leq m}|\omega_j|_{1}$ be a metric on flat torus $\mathbb{T}^{m}=\mathbb{R}^{m}/\...
0
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0answers
86 views

Existence of certain diophantine equations

Given large enough integer $m$ and $3\leq t=O(1)$ integers $m<a_1,a_2,\dots,a_{t-1},a_t<2m$ is there $n\in\Bbb N$ such that there is coprime $n<u,v<c\cdot n$ at fixed $c>1$ with $$ux_i+...
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1answer
147 views

Is it possible to approximate irrational by fractions with denominator and numerator odd? [closed]

Suppose $\alpha$ is a positive irrational, and $\epsilon$ is an arbitrary positive real, are there $m,n$(non-negative integers) such that $$|\alpha-(2m+1)/(2n+1)|<\epsilon/(2n+1)?$$ If they exist, ...
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0answers
81 views

A high dimensional generation of Dirichlet approximation theorem, linear case and nonlinear case

I am working with something on Diophantine approximation, and I found a high dimensional generation of Dirichlet approximation theorem which may be true; I will be very happy if this is true. The ...
4
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0answers
163 views

Asymptotic formula, polynomial, irrational number and uniformly distribution

Problem 1 Given a irrational number $\alpha$ and two polynomials with positive integer coefficients $P(n),Q(n)$, is it possible to get the asymptotic estimate and reasonable error term for: $$\...
-1
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1answer
135 views

Inverse to a Diophantine number [closed]

A number $\alpha$ is said to satisfy the Diophantine condition with exponent $\beta$ iff for some constant $C>0$ the estimate $$ \left| \alpha - \frac{p}{q} \right| > \frac{C}{q^{2+\beta}} $$ ...
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0answers
277 views

Diophantine approximation of algebraic number

An important theorem in Diophantine approximation is the theorem of Liouville: Liouville Theorem If x is a algebraic number of degree $n$ over the rational numbers then there exists a constant c(x) ...
2
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1answer
297 views

Counting algebraic points of bounded height

Let $K$ be a number field and $X\hookrightarrow\mathbb P^n_K$ be a projective variety of degree $\delta$ (with respect to universal bundle) and dimension $d$. We denote the set $$S(X;D,B)=\{\xi\in X(\...
78
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4answers
6k views

Is the series $\sum_n|\sin n|^n/n$ convergent?

Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent? (The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for ...
2
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0answers
55 views

Counting rare events in Diophantine approximation - distributional estimates?

Is there a sort of distributional estimate in Diophantine approximation which allows to estimate the number of solutions which provide a certain quality of approximation? For example, how large is the ...
9
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1answer
217 views

Cover of the positive real numbers by intervals

For which real numbers $x$ and $y$ does the following hold?: $$ \bigcup_{\frac{a}{b} \in \mathbb{Q}^+} \left[\frac{a}{b},\frac{a}{b}+\frac{1}{a^x b^y}\right] \ = \ \mathbb{R}^+ $$
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0answers
109 views

$L^\infty$ norm lower bound for Integer points in null spaces of recursively defined integer vectors?

Letting $\otimes$ be matrix kronecker/tensor product with $n\in\Bbb N$ as a parameter define non-negative integer vectors recursively $$v_1=\begin{bmatrix}a_1&b_1\end{bmatrix}\in\Bbb Z_{>0}^2$$ ...
2
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1answer
77 views

Limits of a quasiperiodic function with two pseudoperiods

Let $\beta$ be a real number such that $\beta^2\notin\mathbb{Q}$. For any smooth function $f$ on $\mathbb{R}$ that decreases sufficiently at infinity, for example a Gaussian function, let us define $$ ...
10
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1answer
189 views

Distribution of good diophantine approximations

Let $\langle x \rangle: \mathbb{R} \to (-1/2,1/2]$ be the periodic function with period $1$ which is $x$ for $x \in (-1/2,1/2]$. Is there some function $D(a,b)$ of real numbers $a<b$ such that, for ...
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0answers
186 views

How large are good approximations to irrational numbers?

It is well known that for almost every $c \in \mathbb{R} / \mathbb{Z}$ there exists $(q_n)_{n \geq 1}$ and $(a_n)_{n \geq 1}$ such that $$|c - a_n / q_n| \leq 1/ q_n^2,$$ where $q_n < q_{n+1} \leq ...
10
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4answers
284 views

Sequential addition of points on a circle, optimizing asymptotic packing radius

Suppose I have to put $N$ points $x_1, x_2, \ldots, x_N$ on the circle $S^1$ of length 1 so as to achieve the largest minimum separation (packing radius). The optimal solution is the equally spaced ...
6
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2answers
387 views

A naive diophantine approximation question

Let $\alpha$ be a positive real number (bigger than one, and irrational if it matters) (the one I am secretly thinking of is $\varphi,$ the golden ratio). I want to know, given an $\epsilon>0,$ ...
8
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2answers
287 views

Growth of a linear recurrent sequence

Consider the sequence defined by $a_0=a_1=1$ and $a_n=2a_{n-1}-3a_{n-2}$ for $n\geq 2$. This is the sequence https://oeis.org/A087455. I would like to prove that $|a_n|>100$ when $n>10$. How ...
2
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0answers
80 views

Zero digits of a p-adic algebraic number

This question might be too simple and I just don't see something very obvious, so I apologize in advance if that is so. Let $p$ be a prime number and let $\mathbb Z_p$ denote the ring of $p$-adic ...
2
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0answers
229 views

What are the best current bounds on $\times a \times b$?

Let $a,b \in \mathbb{N}_{\ge 2}$ be two integers that are multiplicatively independent (i.e., are not powers of the same integer). I have seen (Bourgain, Lindenstrauss, Michel, Venkatesh: Some ...
2
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1answer
163 views

Bounds on near-zero integer linear combinations of numbers linearly independent over $\mathbb{Q}$

Let $\alpha_1, \alpha_2, \dots$ be an infinite sequence of real numbers such that any finite subset is linearly independent over $\mathbb{Q}$. Let $f(N)$ be the number of tuples $(m_1, \dots, m_N)$ ...
9
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1answer
280 views

Simultaneous Diophantine approximation of $\sqrt{2}$ and $\sqrt{2\pm \sqrt{3}}$

By using the LLL algorithm, I tried to find the best simultaneous Diophantine approximation of the three numbers $\sqrt{2} $ and $ \sqrt{2 \pm \sqrt{3}} $. I was expecting that to get a precision of $\...
0
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0answers
107 views

Simultaneous Diophantine approximation in the non-generic case

Suppose we have $n$ irrational numbers $\{ x_1, x_2, \ldots, x_n \}$. For a generic set of such numbers, we have the well-known theorem that there exist infinitely many integers $q$ such that $$ \...
3
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0answers
283 views

On discrepancy of integer sequences related to Erdos-Turan-Koksma

Assume we have a sequence $a_1,\dots,a_k\in\Bbb N$ with each $a_i\approx n$ where $n$ is some integer. Suppose there exists an $\eta\in(0,1)$ such that for every $d_1,\dots,d_k\in\Bbb Z$ with $$...
1
vote
2answers
338 views

Beyond Dirichlet's approximation theorem

I haven't taken any number theory courses but out of curiosity I learned about Dirichlet's approximation theorem. Afterwards, it occurred to me to define the following function $f$ using 'optimal' ...
38
votes
1answer
1k views

Diophantine equation for 2016: triangular $|{\rm GL}_2({\bf F}_q)|$

For a prime power $q$ the group ${\rm GL}_2({\bf F}_q)$ has $(q^2-1)(q^2-q)$ elements. This happens to be a triangular number for $q=2$ (being $6 = 1+2+3$), and $-$ more notably, especially this year ...
12
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2answers
435 views

Are there any solutions to the diophantine equation $x^n-2y^n=1$ with $x>1$ and $n>2$?

This problem arose when considering storage of cannonballs in n-dimensional pirate ships, as explained in this MSE post. This MO question can also be reduced to the $n=3$ case. If $x,y$ is a solution ...