198 votes
Accepted

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

Note that if $\pi$ were rational (with even numerator), then $\sin(n)$ would equal $1$ periodically, so the series would diverge. Similarly if $\pi$ were a sufficiently strong Liouville number. Thus,...
Terry Tao's user avatar
  • 109k
40 votes

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

On the OP request, here is the plot of first 10000 partial sums. Following Terry Tao's suggestion, here is the plot of ($n$th partial sum) $+2^{\frac32}/\sqrt{\pi n}$ for $n$ up to one million: The ...
28 votes
Accepted

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

$$\sqrt{2+\sqrt{3}}-\sqrt{2- \sqrt{3}}=\sqrt{2}$$
Fedor Petrov's user avatar
27 votes

For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

It's indeed unbounded for every irrational $x$. Let me identify points of $\mathbb{R}/\mathbb{Z}$ with their representatives on $[0,1)$, and order it by the usual order $<$ of $\mathbb{R}$ applied ...
Ville Salo's user avatar
  • 6,337
26 votes
Accepted

$\lfloor \frac{3^n}{2^n} \rfloor = \lfloor \frac{3^n-1}{2^n-1} \rfloor$ for $n\geq2$

Mahler proved in 1957 (see here) that if $q$ is a positive rational number which is not an integer, then the distance of $q^n$ to the nearest integer is $(1-o(1))^n$. In particular, taking $q=3/2$, we ...
GH from MO's user avatar
24 votes
Accepted

Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

The product does not tend to the limit zero. For any irrational number $\alpha$ one can show that $$ \limsup_{N\to \infty} \prod_{n=1}^{N} |1- e(n\alpha)| = \infty. \tag{1} $$ (Here I use the usual ...
Lucia's user avatar
  • 43.3k
20 votes
Accepted

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

Delone (1930) and Nagell (1928) showed for any nonzero integer $d$ that the equation $x^3 - dy^3 = 1$ has at most one solution in integers $(x,y)$ besides $(1,0)$, with no constraint on the signs of $...
KConrad's user avatar
  • 49.6k
20 votes

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

Semilog plot building on მამუკა ჯიბლაძე's picture, this time to $10^7$
19 votes

Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

This integral has a series counterpart $$\sum_{k=0}^\infty \frac{240}{(4k+5)(4k+6)(4k+7)(4k+9)(4k+10)(4k+11)}=\frac{22}{7}-\pi$$ https://math.stackexchange.com/a/1657416/134791 (UPDATE Peter Bala ...
Jaume Oliver Lafont's user avatar
19 votes
Accepted

Can two rational rotations $F_2 = \langle A, B \rangle \to SO(3)$ efficiently approximate the $3 \times 3$ identity matrix?

Setting $a_0=A^7, b_0=B^7$ and then $a_{n+1}=[b_n^{-1},a_n], b_{n+1}=[a_n,b_n]$ seems very efficient. The length grows like $C \alpha^n$ with $\alpha= \frac{3 + \sqrt{17}}2$ and the operator norm ...
Andreas Thom's user avatar
  • 25.3k
16 votes

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

Squares of elements of the $3$-adic integers $\mathbb{Z}_3$ are congruent to $0$ or $1$ modulo $3$, thus they are all at $3$-adic distance $1$ from $2$. Squares of elements of $\mathbb{Q}_3 \setminus ...
GNiklasch's user avatar
  • 2,391
16 votes

Digits in an algebraic irrational number

What is known is that every real irrational has a $0$ in its $g$-ary expansion for infinitely many $g$. WLOG take $0 < x < 1$. Taking an even-numbered convergent of the continued fraction of $...
Robert Israel's user avatar
15 votes

Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

I have no idea how to approach the general problem, but here is a quick observation: A. Let $\alpha = 2\beta$ so that $\beta$ is irrational if and only if $\alpha$ is so. Define $f$ by $f(x) = \log|2\...
Sangchul Lee's user avatar
15 votes
Accepted

Upper bound for Hall's conjecture on separation of squares and cubes

The best $\theta$ is $0$. It is known that there are infinitely many solutions of 0 < $|x^3 - y^2| \ll x^{1/2}$, parametrized by certain "Pell equations"; indeed one such family attains $...
Noam D. Elkies's user avatar
14 votes
Accepted

Baker's theorem for integer combinations of logarithms of integers?

The main difficulty in proving Baker's theorem is in estimating $C$. If you don't care about those estimates, then the proof is not difficult. For example, Chapter 7 of Waldschmidt's Diophantine ...
Oleg Eroshkin's user avatar
13 votes

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

Let $D_N$ be the discrepancy: $$ D_N=\sup \left| \frac{ A(J:P)}{N} - |J|\right| $$ where $P=\{k/\pi \ \mathrm{mod} \ 1\}_{k=1,2,\ldots, n}$, $J$ is an interval in $[0,1]$. If the irrationality ...
Sungjin Kim's user avatar
  • 3,310
13 votes

For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

We have Theorem. Let $\psi(x)$ and $\varphi(x)$ be positive increasing functions such that $$\int_1^\infty \frac{dx}{\psi(x)}=+\infty,\qquad \int_1^\infty \frac{dx}{\varphi(x)}<+\infty.$$ Then ...
juan's user avatar
  • 6,976
13 votes
Accepted

Continuous variant of the Chinese remainder theorem

Your guess is correct. For $\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_d) \in \mathbb{R}^d$, the values of $m \alpha \bmod 1$ are equidistributed (and in particular dense) in $(\mathbb{R}/\mathbb{Z})...
David E Speyer's user avatar
12 votes

Estimate number of solutions in the Roth's theorem

For a fixed $\alpha$, the number $N_{\alpha}(\epsilon)$ is bounded by a polynomial function of $1/\epsilon$. The proof of this requires either Faltings's product theorem, or Esnault and Viehweg's ...
Vesselin Dimitrov's user avatar
12 votes
Accepted

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

Here is a full solution for the modified problem, inspired by Gro-Tsen's valuable comment. 1. There are infinitely many rational numbers $a/b\in\mathbb{Q}$ in lowest terms such that $$ \left|\frac{a^...
GH from MO's user avatar
11 votes

Estimate number of solutions in the Roth's theorem

I believe that there is a completely explicit upper bound for $N_\alpha(\epsilon)$ (more generally counting in relative number fields and using more then one, possibly non-archimedean, absolute value) ...
Joe Silverman's user avatar
11 votes

Approximating any integer by multiples of 2 and 3

Fix $\varepsilon>0$. We have to prove (for large enough $n$) that there exist non-negative integers $a,b$ such that $\log_2 n\leqslant a+b\log_2 3<\log_2 n+\varepsilon$. This follows from the ...
Fedor Petrov's user avatar
11 votes

A naive diophantine approximation question

The answer is no. Start with $\alpha$ between 3.25 and 3.75; take squares of those 2 to see that one can restrict $\alpha^2$ to being between 11.25 and 11.75; take 3/2 powers of those to see that one ...
Yaakov Baruch's user avatar
11 votes

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

A non-trivial lower bound can be obtained using linear forms in $p$-adic logarithms. Suppose that $\left\{\left(\frac{4}{3}\right)^n\right\}$ is small. Clearly it is a rational number with denominator ...
Jan-Christoph Schlage-Puchta's user avatar
11 votes
Accepted

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

Yes. Assume that $\alpha$ is irrational, and its continued fraction digits do not exceed $K$. Then, for any positive integer $q$, we have $$q\|q\alpha\|>1/(K+2).$$ In particular, for $q\leq N$, we ...
GH from MO's user avatar
10 votes

Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

Write it as $a_n(\alpha)$ to emphasize the dependence on $\alpha$. For any $\epsilon > 0$, $U(n,\epsilon) = \{\alpha: |a_n(\alpha)| < \epsilon\}$ is an open set containing $k/m$ for any integers ...
Robert Israel's user avatar
10 votes
Accepted

Growth of a linear recurrent sequence

First note that $\frac{1+i\sqrt{2}}{1-i\sqrt{2}}\in\mathbb{Q}(\sqrt{-2})$ is not a root of unity, because it does not equal $\pm 1$. Therefore Baker's famous theorem shows that, for some effectively ...
GH from MO's user avatar
10 votes
Accepted

Sign in Dirichlet's approximation theorem

Yes, this follows from considering the continued fraction of $\alpha$. If $p_n/q_n$ is the $n$th convergent to $\alpha$ and $n$ is odd then $$ 0\leq \alpha - \frac{p_n}{q_n} \leq \frac{1}{q_n^2}.$$
Thomas Bloom's user avatar
  • 6,608
10 votes

For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

Here is an argument essentially due to fedja I learned about thirteen years ago on artofproblemsolving.com. Proposition: if $f$ is $2$-periodic Riemann integrable such that $\sup_{n \geq 1} \left|\...
Paata Ivanishvili's user avatar
10 votes

Well known applications of Roth's theorem

If nothing else we can use Roth's theorem to generalize Liouville's construction of transcendental numbers. Liouville noted that numbers such as $\lambda = \sum_{k=1}^\infty 1 / 10^{k!}$ are ...
Noam D. Elkies's user avatar

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