# Tag Info

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,...
• 109k

### 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 ...
Accepted

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

$$\sqrt{2+\sqrt{3}}-\sqrt{2- \sqrt{3}}=\sqrt{2}$$
• 103k

### 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 ...
• 6,337
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 ...
• 99k
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 ...
• 43.3k
Accepted

• 2,391

• 564
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• 151k

### 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 ...
• 13.7k
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• 6,608