All Questions
Tagged with ds.dynamical-systems nt.number-theory
45 questions with no upvoted or accepted answers
57
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On the first sequence without triple in arithmetic progression
In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
23
votes
0
answers
896
views
Base change for $\sqrt{2}.$
This is a direct follow-up to Conjecture on irrational algebraic numbers.
Take the decimal expansion for $\sqrt{2},$ but now think of it as the base $11$ expansion of some number $\theta_{11}.$ Is ...
- 96.4k
18
votes
0
answers
667
views
The lonely molecule
Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with $n=...
- 151k
13
votes
0
answers
802
views
Hilbert 16th problem and dynamical Lefschetz trace formula
I would like to apply the known version of the conjectural formula (11) page 10 of the paper Number theory and dynamical Lefschetz trace formula.
Disclaimer: I do not have a complete ...
- 356
11
votes
0
answers
809
views
Borderline Collatz-like problems
The usual Collatz map is $C:n \mapsto n/2$ if $n$ even, $(3n+1)/2$ if $n$ odd. Let $f^{\circ (r+1)}:=f \circ f^{\circ r}$.
We suspect that for every fixed $n>0$, the sequence $C^{\circ r}(n)$ ...
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
11
votes
0
answers
853
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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
9
votes
0
answers
225
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On the first sequence without collinear triple
Let $u_n$ be the sequence lexicographically first among the sequences of nonnegative integers with graphs without collinear three points (as for $a_n=n^2$ or $b_n=2^n$). It is a variation of that one.
...
9
votes
0
answers
285
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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:
$$ (x,y,z) \mapsto \text{sort}...
- 22.8k
8
votes
0
answers
197
views
The condition on $\alpha$ that $\alpha^n$ is convergent modulo 1
We consider numbers $\alpha\in \mathbb{R}$ with $|\alpha|>1$. Is there any result about a characterization of those $\alpha$ so that $\{\alpha^n\}_{n\in \mathbb{N}}$ is convergent modulo 1?
I ...
- 661
7
votes
0
answers
165
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Lonely globe trotters
In analogy with the lonely runners conjecture,
imagine "globe trotters" each traveling on a longitudinal great circle at different
(constant, positive) speeds. Each "trotter" ...
- 151k
7
votes
0
answers
429
views
Dynamics of a curious bijection of $\mathbb N$
The two sequences A48680 and A48679 of the OEIS define two mutually inverse bijections on the set of all strictly positive natural numbers given (for the comfort of the reader) as follows:
Given an ...
- 17.9k
7
votes
0
answers
156
views
Partition the rationals with respect to a multivariate polynomial which sends classes to classes
Let $R$ be a commutative ring and let $f\in R[x_1,x_2,\cdots,x_{n-1}],n\geq 2$ be a polynomial.
Definition: We say $f$ is $n$-severable over $R$ if there exists a partition (of set) $$R=\coprod_{i=...
- 171
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
6
votes
0
answers
448
views
Are there always at least *five* divisions?
@JosephO'Rourke asked a question about a Collatz like function related to primes:
$f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is prime} \\
\lfloor n/2 \rfloor & \text{if} \;n \;\text{...
- 1,375
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
5
votes
0
answers
282
views
Limits of $p/\ln p - q /\ln q$, $p, q$ prime
Is there any $\alpha>0$ for which there are known to exist two sequences of primes, $(p_i), (q_i)$ such that
$$\alpha = \lim_{i\to\infty} \left(p_i/\ln p_i - q_i /\ln q_i\right)\ ?$$
The ...
- 17.6k
4
votes
1
answer
373
views
Equidistribution of the orbit $\{\text{diag}(t^a,t^{-a})\Lambda \}_{t>0}$ for a.e. $\Lambda\in \text{SL}(2,\mathbb R)/\text{SL}(2,\mathbb Z)$
$\DeclareMathOperator\diag{diag}\DeclareMathOperator\SL{SL}$It is well-known that geodesic flow $g_t=\{\diag(e^t,e^{-t}) \}_{t>0}$ acts ergodically (actually mixing) on $\SL(2,\mathbb R)$ (Howe–...
- 321
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:
$$\...
- 697
4
votes
0
answers
96
views
On decidability of an infinite dimensional dynamic system
Consider two infinite dimensional Toeplitz integer matrices $A,B$ where $B$ is just a shift operator and an infinite dimensional vector $V_0$.
Given a prime $p$ and an infinite dimensional integer ...
- 13.9k
3
votes
0
answers
271
views
Approximating rotations on a torus with irrational rotations
Consider a rotation of the form $x\mapsto e^{i\theta}x$, for $x$ on the unit circle. By iterating this rotation, one can approximate any other rotation $x\mapsto e^{i\phi}x$ arbitrarily well, as long ...
- 181
3
votes
0
answers
211
views
uniform bounds on Weyl Equidistribution theorem?
If $\theta \notin \mathbb{Q}$ the sequence $\{ n \theta\}$ is equidstributed mod 1. If we let $f \in L^2 ([0,1])$ and $T: x \mapsto x + \theta $ this could be phrased a special case of the ergodic ...
- 22.8k
3
votes
0
answers
309
views
A Dedekind Eta trajectory / horocyclic flow (Reference request)
I've been exploring the composition of essentially the Dedekind $\eta$-function with
parabolic Möbius transformations,
$$C_L(z,t)=\left(\frac{z}{-tz+1}\right)^{\frac{1}{2}}\eta\left(\frac{z}{-tz+1}\...
- 10.5k
2
votes
0
answers
61
views
Aligning frequencies
Let $\omega_1, \omega_2, \dots, \omega_n$ be frequencies between $1$ and $\log n$. I would like to find an upper bound for a point $t$ that align these frequencies up to a small error $\delta$, that ...
- 21
2
votes
0
answers
137
views
Entropy of a sequence
I am reading the paper Sign Changes in Hecke Eigenvalues by Matomaki and Radziwill, and in one place they mention the following,
It would be interesting to rule out the possibility of $\lambda_f(n)$ ...
- 589
2
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0
answers
281
views
Surreal numbers and the Collatz iteration as a game?
Let us define a game based on the Collatz function $C(n) = n/2$ if $n$ is even, otherwise $=3n+1$.
Each number $n$ represents a game played by left $L$ and right $R$:
$$n = \{L_n | R_n \}$$
The rules ...
- 3,064
2
votes
0
answers
91
views
Does there exist a natural number $m$ such that $\sigma^{(k)}((2m+1)^2)$ is an odd square number for all $k\ge 0$?
This question, comes out of a question in MSE and I hope it is ok to ask it here:
Does there exist a natural number $m$ such that $\sigma^{(k)}((2m+1)^2)$ is an odd square number for all $k\ge 0$?
...
- 3,064
2
votes
0
answers
125
views
Does $\sum_{i\le k}\mathrm{frac}(n\alpha_i)<1$ hold infinitely often?
For each $t \in \mathbf{R}$, let $\mathrm{frac}(t)$ be its fractional part.
Question. Fix reals $\alpha_1,\ldots,\alpha_k \in (0,1)$ such that $\sum_{i\le k}\alpha_i<1$. Do there exist ...
- 1,511
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
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
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
2
votes
0
answers
248
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 \mathbb{R}^...
- 616
2
votes
0
answers
118
views
random maass waveforms
Let $H$ be the upper half complex plane and $\Gamma$ a discrete subgroup of $SL_2(\mathbb{Z})$ such that the volume of of $\Gamma \backslash H$ is finite. There is a conjecture of Berry that Maass ...
- 101
1
vote
0
answers
84
views
Coarse well-distributedness/equidistribution of Pell sequence prefixes
I am interested in the distributedness or "mixing" behavior of certain
linear recurrences modulo powers of $2$.
In particular, consider the Pell sequence (https://oeis.org/A000129),
modulo $...
- 11
1
vote
0
answers
193
views
Theoretical invariant distribution of discrete dynamical systems, including the Riemann Zeta map
Update on 3/10/2021: I added Example 5 in the Appendix. This generic example encompasses the Riemann Zeta dynamical system. A simple version of this post, targeted to engineers, machine learning ...
- 3,098
1
vote
0
answers
342
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Are the orbits of this discrete dynamical system bounded?
Somehow I believe this should be true and easy to prove but cannot nail it down. A reference, proof, or counterexample will suffice. Didn't get any help over at MSE even with a bounty so I came here.
...
- 143
1
vote
0
answers
87
views
How fast will it converge to equilibrium？
$\alpha \in \mathbb{R} \backslash \mathbb{Q}$, given $\varepsilon>0 \quad \lambda>0$.
given $n_{1}, \cdots, n_{k} \in N^{*}$ satisfied $(1-\varepsilon) \lambda<n_{1}<\cdots<n_{k}<(1+...
- 543
1
vote
0
answers
138
views
Diophantine approximation and the Euclidean algorithm
My question is whether something I've noticed is well-known. It seems like it must be, but I've been unable to find any references that describe what is outlined below.
Given real $x$ and irrational $...
- 133
1
vote
0
answers
271
views
Is my ansatz for finding $n$-periodic-points of the exponential-function exhaustive?
The following is about getting help for a proof on existence and indexability of periodic points of the exponential-function, here with base $e:=\exp(1)$.
Update The question is a complete rewriting ...
- 5,342
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
1
vote
0
answers
122
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square tiled surfaces: Counting Saddle Connections vs Counting Square-Tiled Surfaces
I am reading Prime arithmetic Teichmuller discs in H(2) where they discuss the Siegel-Veech constant in a stratum $\mathcal{H}(2)$ of surfaces. However I see two definitions of Siegel-Veech constant:
...
- 22.8k
0
votes
0
answers
81
views
Replacing the sequence in Chowla's conjecture and positiveness of the entropy
For any fixed integer $m>0$ and not-all-even $(a_1,\ldots,a_m)\in\mathbb N^m$, one version of Chowla's conjecture states that
$$
\lim_{x\rightarrow\infty}\frac{1}{x}\sum_{n\leq x}\mu(n+1)^{a_1}\...
- 457
0
votes
0
answers
620
views
Cocycles and the Collatz problem?
Let $T(n) = n+R(n)$, where $R(n) = -n/2 $ if $n\equiv 0 \mod 2$ else $R(n) = \frac{n+1}{2}$.
$R(n)$ is the Cantor ordering of the integers:
https://oeis.org/A001057
In the Collatz problem, one is ...
- 3,064
0
votes
0
answers
89
views
Exact approximation in $p$ adic
Given a non increasing function $\psi$ the $\psi$ approximable points in $\mathbb{R}^n$ is defined as
$W(\psi)=\{x\in\mathbb{R}^n:|qx-p|<\psi(q)\}$ for infinitely many $(q,p)\in \mathbb{Z}^m\times\...
- 11
0
votes
0
answers
96
views
$||g_n||_{\infty} < \delta_{n-1}(g)$
It may be a simple question to post it here, but I posted this question in the Math Stack Exchange forum and no one answered me.
Let $E$ be a (possibly infinite) alphabet and consider $X = E^{\...
- 151