All Questions
18 questions
26
votes
4
answers
2k
views
For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?
For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded?
I feel that it is not easy to treat every irrational $x$.
I have asked in S.E. ...
- 385
18
votes
1
answer
653
views
Can the expansion of a large integer in all bases consist of almost all zeroes?
Let $n$ be a positive integer. Given an integer base $b\ge 2$, let $C_b(n)$ be the number of non-zero digits in the expansion of $N$ in base $b$. Further, let $M(n)=\max\{C_b(n):b\ge 2\}$ be the ...
- 4,660
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:
$$ A = \left[\begin{array}{rcc} \frac{3}{5} & \frac{4}{5} & 0 \\
-\frac{4}{5} & \frac{3}{5} & 0 \\
0 & 0 & 1 \end{array}\right]
\quad\...
- 22.8k
11
votes
1
answer
727
views
A weakening of the Littlewood conjecture
For real numbers $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. Define a function $\ell:\mathbb{R}^2\rightarrow\mathbb{R}$ by
$$\ell(\alpha,\beta)=\liminf_{n\rightarrow\infty}n\|...
- 1,723
9
votes
3
answers
3k
views
Simultaneous diophantine approximation
Let $r(x)$ be the function $x$ mod $1$, i.e. $x$ minus its floor.
Now let $m$ be a given positive integer, and $c$ a vector in $\mathbb{R}^m$ whose components are linearly independent over $\mathbb{Q}...
- 409
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
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
6
votes
1
answer
465
views
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 ...
- 61
4
votes
2
answers
602
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 ...
- 43
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:
$$\...
- 697
2
votes
0
answers
61
views
Aligning frequencies
Let $\omega_1, \omega_2, \dots, \omega_n$ be frequencies between $1$ and $\log n$. I would like to find an upper bound for a point $t$ that align these frequencies up to a small error $\delta$, that ...
- 21
2
votes
0
answers
125
views
Does $\sum_{i\le k}\mathrm{frac}(n\alpha_i)<1$ hold infinitely often?
For each $t \in \mathbf{R}$, let $\mathrm{frac}(t)$ be its fractional part.
Question. Fix reals $\alpha_1,\ldots,\alpha_k \in (0,1)$ such that $\sum_{i\le k}\alpha_i<1$. Do there exist ...
- 1,511
2
votes
0
answers
261
views
What are the best current bounds on $\times a \times b$?
Let $a,b \in \mathbb{N}_{\ge 2}$ be two integers that are multiplicatively independent (i.e., are not powers of the same integer). I have seen (Bourgain, Lindenstrauss, Michel, Venkatesh: Some ...
- 22.8k
2
votes
0
answers
248
views
Linear forms with best approximation vectors lying in a subspace
Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in \mathbb{R}^...
- 616
1
vote
1
answer
235
views
The liminf of an expression involving an irrational rotation
Let $0 < a < 1$ be an irrational number. Is it true that
$$\liminf_{n \in \mathbb N, n \to \infty} n \{na\} = 0?$$
Note: Here $\{\cdot\}$ denotes the fractional part.
- 6,155
1
vote
0
answers
138
views
Diophantine approximation and the Euclidean algorithm
My question is whether something I've noticed is well-known. It seems like it must be, but I've been unable to find any references that describe what is outlined below.
Given real $x$ and irrational $...
- 133
0
votes
0
answers
89
views
Exact approximation in $p$ adic
Given a non increasing function $\psi$ the $\psi$ approximable points in $\mathbb{R}^n$ is defined as
$W(\psi)=\{x\in\mathbb{R}^n:|qx-p|<\psi(q)\}$ for infinitely many $(q,p)\in \mathbb{Z}^m\times\...
- 11
-5
votes
1
answer
592
views
Central limit theorem for irrational rotations
Let $\alpha$ be an algebraic integer of modulus 1, and $ R_\alpha z=\alpha z$. Is
$$\lim_{n\to\infty}\frac{\log|\sum_{k=1}^n \Re R_\alpha^k z|}{\log n}=\frac12$$ for all $z\in S^1$?
Birkhoff's ergodic ...
- 2,135