The present paper shows that for any $s$ sequences of real numbers, each with infinitely many distinct elements, ${\lambda }_n^j$, j=1,...,s, the rational combinations of $x_1^{{\lambda }_{m_1}^1}{x}_2^{{\lambda }_{m_2}^2}...x_s^{{\lambda }_{m_s}^s}$ are always dense in $C_{I^s}$.

We investigate in a geometrical way the point sets of  $ℝ$  obtained by the  $\beta$-numeration that are the  $\beta$-integers  ${\mathbb{Z}}_{\beta }\subset \mathbb{Z}\left[\beta \right]$  where  $\beta$  is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the  $\beta$-numeration, allowing to lift up the  $\beta$-integers to some points of the lattice  ${\mathbb{Z}}^m$  ($m=$  degree of  $\beta$) lying about the dominant eigenspace of the companion matrix of  $\beta$ . When  $\beta$  is in particular a Pisot number, this framework gives another proof of the fact...

We establish two truncations of Gauss’ square exponent theorem and a finite extension of Euler’s identity. For instance, we prove that for any positive integer $n$, $$\sum _{k=0}^n{(-1)}^k\left[\begin{array}{c}2n-k\\ k\end{array}\right]{(q;q^2)}_{n-k}{q}^{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)}=\sum _{k=-n}^n{(-1)}^k{q}^{k^2},$$ where $$\left[\begin{array}{c}n\\ m\end{array}\right]=\prod _{k=1}^m\frac{1-q^{n-k+1}}{1-q^k}\text{and}{(a;q)}_n=\prod _{k=0}^{n-1}(1-a{q}^k).$$

It is proved that for any $f,g\in A\left(S\right)$, where $S$ is the Poulsen simplex, so that $f,g$ and $1-f$, $1-g$ are smooth points, there is a rotation of $A\left(S\right)$ carrying $f$ in $g$.

It is proved that if ${\phi }_n$ is a complete orthonormal system of bounded functions and ɛ>0, then there exists a measurable set E ⊂ [0,1] with measure |E|>1-ɛ, a measurable function μ(x), 0 < μ(x) ≤ 1, μ(x) ≡ 1 on E, and a series of the form ${\sum }_{k=1}^{\infty }{c}_k{\phi }_k\left(x\right)$, where $c_k\in l_q$ for all q>2, with the following properties: 1. For any p ∈ [1,2) and $f\in L_{\mu }^p[0,1]=f:{ʃ}_0^1{\left|f\left(x\right)\right|}^p\mu \left(x\right)dx<\infty$ there are numbers ${\varepsilon }_k$, k=1,2,…, ${\varepsilon }_k$ = 1 or 0, such that $li{m}_{n\to \infty }{ʃ}_0^1{|{\sum }_{k=1}^n{\varepsilon }_k{c}_k{\phi }_k\left(x\right)-f\left(x\right)|}^p\mu \left(x\right)dx=0.$ 2. For every p ∈ [1,2) and $f\in L_{\mu }^p[0,1]$ there are a function $g\in L^1[0,1]$ with g(x) = f(x) on E and numbers ${\delta }_k$, k=1,2,…, ${\delta }_k=1$ or 0,...