Is there an "elementary" (say ultrafilter-free) proof of the following fact: if $G$ is a compact (Hausdorff) topological group, if $g \in G$ is any element from this group, and if $P$ is a polynomial with integer coefficients without constant term, then the identity element of $G$ is a limit point of the sequence $n \mapsto g^{P(n)}$.


An other question: for which integer-valued sequences $u\_n$ is the result above still true with $P(n)$ replaced by $u\_n$, whatever $G$ and $g$ are?



The identity element of a compact group is a limit point of any "polynomial sequence"
If $\mathcal{H}$ is a van der Corput set of positive integers, then the closure of $\{ g^h\mid h\in\mathcal{H} \}$ contains the identity element. This generalizes the statement in the original post, because $\{ P(n)\mid n>0 \}$ is a van der Corput set for $P\in\mathbb{Z}[x]$ without a constant term. 


By Terry Tao's remark above, it suffices to show that for any $0<\epsilon\leq 1/2$ and for any $\theta\_1,\dots,\theta\_n\in\mathbb{R}$ there is $h\in\mathcal{H}$ such that $\|h\theta\_1\|,\dots,\|h\theta\_n\|<\epsilon$. We follow closely the proof of Theorem 9 in Chapter 2 of Montgomery: Ten lectures on the interface between analytic number theory and harmonic analysis. By the earlier results in the chapter, there is a cosine polynomial
$$ T(x) = a\_0 + \sum\_{h\in\mathcal{H}} a\_h \cos 2\pi hx $$
with real coefficients, nonnegative values, $a\_0<\epsilon^n$, and $T(0)=1$. Put 
$$ f(x):=\max(0,1-\|x\|/\epsilon), $$
and consider the expression
$$ a\_0 + \sum\_{h\in\mathcal{H}} a\_h f(h\theta\_1)\dots f(h\theta\_n). $$
It suffices to show that this expression exceeds $\epsilon^n$, because then $f(h\theta\_1)\dots f(h\theta\_n)\neq 0$ follows for some $h\in\mathcal{H}$. The Fourier expansion
$$ f(x) = \sum\_{k\in\mathbb{Z}}\hat f(k) e(kx) $$
converges absolutely, hence the above expression equals
$$ \sum\_{h\in\mathcal{H}\cup\{0\}} a\_h 
\left(\sum\_{k\_1\in\mathbb{Z}}\hat f(k\_1) e(hk\_1\theta\_1)\right)\dots \left(\sum\_{k\_n\in\mathbb{Z}}\hat f(k\_n) e(hk\_n\theta\_n)\right)=$$ 
$$\sum\_{k\_1,\dots, k\_n\in\mathbb{Z}}\hat f(k\_1)\dots\hat f(k\_n)
\sum\_{h\in\mathcal{H}\cup\{0\}} a\_h e(hk\_1\theta\_1+\dots+hk\_n\theta\_n)=$$
$$\sum\_{k\_1,\dots, k\_n\in\mathbb{Z}}\hat f(k\_1)\dots\hat f(k\_n)
\sum\_{h\in\mathcal{H}\cup\{0\}} a\_h \cos 2\pi h(k\_1\theta\_1+\dots+k\_n\theta\_n)=$$
$$\sum\_{k\_1,\dots, k\_n\in\mathbb{Z}}\hat f(k\_1)\dots\hat f(k\_n)T(k\_1\theta\_1+\dots+k\_n\theta\_n).$$
The term corresponding to $k\_1=\dots=k\_n=0$ contributes $\epsilon^n$, while all the other terms are nonnegative, hence we are done.


