All Questions
Tagged with ds.dynamical-systems nt.number-theory
140 questions
11
votes
0
answers
252
views
Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?
Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...
- 1,914
9
votes
1
answer
255
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 $c$) ...
- 109k
2
votes
0
answers
137
views
Can this construction generate bounded aperiodic functions?
This question is based on this old MathOverflow question: How this set of functions is ordered?
In that question, Vladimir Reshetnikov asked about a class $S$ of functions $f:\mathbb{N}\to\mathbb{N}$ ...
- 2,585
7
votes
1
answer
333
views
Can we estimate the error $\left| \frac{1}{N^2} \sum f ( \{ \sqrt{2} m + \sqrt{3} n \} ) - \int_0^1 f(x) \, dx \right|$?
As a computer experiment I did a few Riemannian sums to see if I could quantify the density statement $\overline{\mathbb{Q}(\sqrt{2}, \sqrt{3})} = \mathbb{R}$ :
$$ \Big| \frac{1}{N^2} \sum_{0 \leq m,...
- 22.8k
7
votes
0
answers
156
views
Partition the rationals with respect to a multivariate polynomial which sends classes to classes
Let $R$ be a commutative ring and let $f\in R[x_1,x_2,\cdots,x_{n-1}],n\geq 2$ be a polynomial.
Definition: We say $f$ is $n$-severable over $R$ if there exists a partition (of set) $$R=\coprod_{i=...
- 171
6
votes
1
answer
562
views
How can I catalog these generalized Collatz problems?
The Collatz conjecture can be expressed in terms of a ruleset in the language $\{x,+,1,\rightarrow,;\}$:
$x + x + 1 \rightarrow x+x+x+1+1;$
$x + x \rightarrow x;$
Whenever a number matches the LHS ...
- 2,302
9
votes
1
answer
2k
views
A problem involving the inverse Collatz map
Let $C$ be the Collatz map on the natural numbers, defined by:
$$C(n) :=
\begin{cases}
n/2 & \text{if} \;n \;\text{even} \\
(3n+1)/2 & \text{if} \;n \;\text{odd}
\end{cases}$$
The inverse ...
4
votes
0
answers
187
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:
$$\...
- 697
7
votes
1
answer
424
views
Naturally occuring counting process with a 1/log asymptotics?
Besides prime numbers, is there another physically realizable counting process that exhibits a 1/log density ? The reason I am posting this question is that we are measuring the response of a quantum ...
- 79
5
votes
1
answer
188
views
Example of a non-arithmetic Veech surface (other than regular polygon)?
I am reading this paper of Avila and Delecroix of the billiard flow on polygonal surfaces, but I have to get through some basic definitions first. What is a non-arithmetic Veech surface?
A Veech ...
- 22.8k
11
votes
1
answer
433
views
Upper bounds for lattice points in orbits, and representations of binary quadratic forms
Write $\mathbb{Z}^{a\times b}$ for the $a\times b$ integer matrices. Let $n\geq 3$ and $Q\in\mathbb{Z}^{n\times n}$. Let $G=O(Q)$ be the orthogonal group of $Q$. For $X_0\in \mathbb{Z}^{2\times n}$, ...
10
votes
2
answers
678
views
Irrational rotation - recurrence times
I consider the irrational rotation
$T_\alpha(x) = x + \alpha \text{ mod } 1$ for given irrational $\alpha \in [0,1]$. For a given open interval $A \subset [0,1]$ with length $|A|>0$, I consider the ...
- 103
1
vote
1
answer
502
views
A Zsigmondy-theorem-analogy in the generalized Collatz-problem $3x+\rho$?
Remark : I've found a rather trivial answer for this question and so very likely the premise of paralleling it with the Zsigmondy-theorem is wrong, so this question might better be retracted. I'll ...
- 5,342
9
votes
1
answer
374
views
Integrality of iterates of rational functions
Let $f(x)$ be a rational function which is a ratio of two integral polynomials, and $n \in \mathbb Z$. Then the sequence of iterates $n, f(n), f(f(n)), f(f(f(n)), ...$ will be an infinite sequence of ...
- 6,039
32
votes
2
answers
2k
views
A Collatz-like problem on prime numbers
Consider the function $f$ on the prime numbers defined by $$ f(p):= \text{ the greatest prime factor of } 2p+1.$$ The iteration of $f$ from any prime $p<10^8$ converges to the cycle $$(3,7,5,11,23,...
9
votes
2
answers
554
views
The mean value of $y \log{y}$ over the ordinates of the CM points
Let $-D < 0$ be a negative fundamental discriminant and let $y$ range over the values $y = y_Q = \frac{\sqrt{|D|}}{2a}$, as the values $(a,b,c)$ run through the reduced binary quadratic forms $Q = ...
- 13.8k
3
votes
1
answer
395
views
Waldspurger Formula as a Torus Integral
I have a research-level but not necessarily new question about certain equidistribution problems. If $\phi \in L^2(S^2)$ then we could define the Weyl sums:
$$ \int \phi \, \mu_d = \frac{1}{|\mathcal{...
- 22.8k
2
votes
0
answers
261
views
What are the best current bounds on $\times a \times b$?
Let $a,b \in \mathbb{N}_{\ge 2}$ be two integers that are multiplicatively independent (i.e., are not powers of the same integer). I have seen (Bourgain, Lindenstrauss, Michel, Venkatesh: Some ...
- 22.8k
15
votes
2
answers
648
views
Is the following series consisting of equally distributed $\pm 1$ bounded?
Apologise in advance if this problem isn't research-level (I'm quite certain it isn't). It's just I found it quite intriguing because it turned out to be much more subtle than it appeared at my first ...
- 253
2
votes
3
answers
467
views
Concise introduction to Beta transformations
I'm looking for a concise introductory on the subject of beta transformations on the circle. I've found things that are related to its applications in the computational field or in number theory. A ...
- 3,574
4
votes
0
answers
96
views
On decidability of an infinite dimensional dynamic system
Consider two infinite dimensional Toeplitz integer matrices $A,B$ where $B$ is just a shift operator and an infinite dimensional vector $V_0$.
Given a prime $p$ and an infinite dimensional integer ...
- 13.9k
6
votes
2
answers
473
views
rate of equidistribution of the horocycle flow for $SL(2, \mathbb{Z})$
I know that for any Fuchsian group $\Gamma$, there is a spectral gap, which leads to
$$ \left| \int_0^1 F(x + iy) \, dx - \int_{\Gamma \backslash \mathbb{H}} F \, \frac{dx \, dy}{y^2} \right| < ...
- 22.8k
-2
votes
1
answer
459
views
Find $x,y,z \in \mathbb{Z}$ with $|x^2 + y^2 - \sqrt{3} z^2| < 10^{-6}$ [closed]
does Oppenheim conjecture hold for specific quadratic forms? or for generic quadratic forms with a set of measure 1.
for example can we find $x,y,z \in \mathbb{Z}$ with
$$|x^2 + y^2 - \sqrt{3} z^2| &...
- 22.8k
5
votes
2
answers
341
views
a modification on an infinite Bernoulli convolution
The distribution $\nu_{\lambda}$ of the random series $\sum\pm\lambda^n$ is the infinite convolution product of $\frac12(\delta_{-\lambda^n}+\delta_{\lambda^n})$. This problem has been studied ...
- 43.2k
9
votes
0
answers
285
views
when is the Brun continued fraction periodic?
I was hoping to figure this one out on my own. There's this nice paper by Avila on various "subtractive" Euclidean algorithms. Here is one he attributes to Viggo Brun:
$$ (x,y,z) \mapsto \text{sort}...
- 22.8k
10
votes
2
answers
371
views
Refined equidistribution for the periodic trajectories of Anosov flows?
Duke, and Linnik before him under a restrictive condition, proved that the set of closed geodesics of a given length $L$ is equidistributed on the modular surface as $L \to \infty$. This is a theorem ...
- 13.8k
3
votes
0
answers
211
views
uniform bounds on Weyl Equidistribution theorem?
If $\theta \notin \mathbb{Q}$ the sequence $\{ n \theta\}$ is equidstributed mod 1. If we let $f \in L^2 ([0,1])$ and $T: x \mapsto x + \theta $ this could be phrased a special case of the ergodic ...
- 22.8k
7
votes
0
answers
221
views
integrality of a Riccati-type equation
The following is a problem we were unable to prove and left stated in the paper
"Arithmetical properties of a sequence arising from an arctangent sum", J. Numb. Theory 128 (2008) 1807–1846.
Define ...
- 43.2k
2
votes
1
answer
515
views
On comparing two almost injective divisor maps
Edit 2018.08.08 This answer https://mathoverflow.net/a/307881 will be updated to give recent information about S, especially a forthcoming preprint. End Edit 2018.08.08
In an introductory post on ...
- 13k
5
votes
0
answers
772
views
The Grimm Machine(s): A Collatz Conjecture Rival?
Edit 2018.08.08 This answer https://mathoverflow.net/a/307881 will be updated to give recent information about S, especially a forthcoming preprint. End Edit 2018.08.08
Just as the Collatz ...
- 13k
4
votes
1
answer
677
views
Are the Farey numbers $\{ \frac{a}{b} : a < b, \gcd(a,b) = 1\}$ uniform in $[0,1]$?
I did some numerical experiments about $\{ \frac{a}{b}: a,b \in A, \;a < \, b\}$ for various integer sets $A$. Does anyone recognize these densities?
Here is $A = \{ p: \text{prime}\}$
Here is $...
- 22.8k
2
votes
2
answers
395
views
About consecutive integers covered by arithmetic progressions
Help me please to solve the following problem.
There are $n$ arithmetic progressions of the form:
$$(2i+1)k + x_i,~~~~ i = 1,\ldots,n, k \geq 0$$
Initial integer terms $x_i \geq 0$ are varying.
...
- 267
22
votes
2
answers
1k
views
$x_1 = 2$, $x_{n + 1} = {{x_n(x_n + 1)}\over2}$, what can we say about $x_n \text{ mod }2$?
This question was asked on MathStackexchange here, but there was no answer, so I am asking it here.
Let$$x_1 = 2, \quad x_{n + 1} = {{x_n(x_n + 1)}\over2}.$$What can we say about the behavior of $x_n ...
user61522
5
votes
4
answers
2k
views
How do these primes jump?
Update 2017.08.28: I am still looking for references. I have posted a request to https://cs.stackexchange.com/q/79971 which includes some literature references I found which are of interest but still ...
- 13k
11
votes
0
answers
853
views
horocycle flow and the prime number theorem
Looking at Zagier's Eisenstein Series and the Riemann Zeta Function, we get a proof of the prime number theorem using horocycles. I would really love it if there were a geometric proof like this.
...
- 22.8k
1
vote
0
answers
151
views
What interesting information can be deduced from knowledge of how deep a geodesic ventures into the cusp
First of all I have to apologise as I am not a geometer and my knowledge of geometry is poor. Let $M$ be the modular surface and $\gamma$ to denote a geodesic in $M$. In the the following paper by ...
- 326
11
votes
1
answer
925
views
About positive upper density
For $S\subset \mathbb{N}$ define the upper density as $D^{\ast
}(S)=\limsup_{n\rightarrow \infty }\frac{\left\vert S\cap \{1,2,\ldots,n\} \right\vert }{%
\left\vert n\right\vert }.$
Question: ...
- 121
3
votes
1
answer
318
views
theta functions and Brownian motion
I did some plots of the theta function $\theta(z) = \sum q^{n^2}$ near the real axis, so $q = e^{2\pi i \, n z}$ and $z = 0.001 + i \mathbb{R}$. At first it looks like some random sine curve and then ...
- 22.8k
4
votes
2
answers
136
views
Sturmian subword whose reverse is not a subword
Let ${\cal L}_n$ be the set of all subwords of length $n$ of a biinfinite Sturmian sequence, induced by a rotation coding with irrational angle $\theta$.
Take a word $w \in {\cal L}_{2^n}$ and write ...
- 2,560
11
votes
2
answers
478
views
Can two rational rotations $F_2 = \langle A, B \rangle \to SO(3)$ efficiently approximate the $3 \times 3$ identity matrix?
Let $A,B$ be two rational rotations:
$$ A = \left[\begin{array}{rcc} \frac{3}{5} & \frac{4}{5} & 0 \\
-\frac{4}{5} & \frac{3}{5} & 0 \\
0 & 0 & 1 \end{array}\right]
\quad\...
- 22.8k
2
votes
0
answers
299
views
A weighted ergodic average
According to my simulations, it looks like the number of times that the $N$ first iterates $u_0$, $\ldots$, $u_{N-1}$ of the sequence $(u_n)$ defined here meets an interval $I$ is close to $N|I|$ ...
- 2,560
1
vote
1
answer
435
views
A quantitative Kronecker theorem
I encounter the following question.
$\textbf{Problem}$: For almost all Matrix $M\in\mathcal M_{m\times n}(\mathbb R),$ all $y\in \mathbb R^m$ and any $N$, small $\epsilon>0$, there exists a ...
- 197
12
votes
1
answer
374
views
Unusual digit sets that allow finite expansions for all (positive and negative) integers
Informal introduction
(If you don't like informal introductions, please skip to 'Mathematical formulation')
Whenever our 'decimal positional system' for writing numbers comes up in conversation, ...
- 2,493
7
votes
1
answer
927
views
Algebraic dynamics in finite fields
What is known about combinatorial structure of the rational maps of degree 2 over finite fields? From some general reasons I think it was studied. For being more specific, consider the field $\mathbb{...
- 109k
22
votes
3
answers
1k
views
Distribution of the Error term in GH Hardy's "curious result" $\sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$
In an early paper, GH Hardy talks about the distribution of "curious" sum:
$$ \sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$$
where $\{x\}:=x-\left \lfloor x \right \rfloor -1/2$. ...
- 22.8k
7
votes
2
answers
321
views
Random suborbits of a rotation
Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...
- 2,560
11
votes
1
answer
727
views
A weakening of the Littlewood conjecture
For real numbers $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. Define a function $\ell:\mathbb{R}^2\rightarrow\mathbb{R}$ by
$$\ell(\alpha,\beta)=\liminf_{n\rightarrow\infty}n\|...
- 1,723
5
votes
2
answers
571
views
Equidistribution of Hecke points and $p = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$
I have seen two versions of a result called "Hecke Equidistribution" and I wanted to know if they were the same or different.
#1 Let $p = 4k+1 = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$. Then $\...
- 22.8k
2
votes
1
answer
433
views
How to find representatives of $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$
While reading about the Teichmuller flow, I am reading about the space of lattices $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$.
I could not a find a good way of computing the Teichmuller flow on this ...
- 22.8k
8
votes
2
answers
480
views
An algorithm for Poincare recurrence time
Define the function $[0,+\infty) \rightarrow R$:
$$ f = \cos (t) + \cos (\sqrt{2} t) + \cos (\sqrt{3} t) + \cos (\sqrt{5} t ) . $$
I want a number $t $ bigger than $10^7$ such that
$$ f(t) > 4 -...
- 193