Given a subset $E\subset \mathbb N\setminus \{0\}$ of integers, we say that $E$ is independent if for any choice of $k\in \mathbb N$, $\epsilon_i\in\{-1,0,1\}$ for $i=1,\dots,k$, if $\Sigma_{i=1}^k\epsilon_in_i=0$ for $n_i\in E$, then $\epsilon_i=0$ for any $i$.

My Question is the following. Is the set $\{5^n3^m\}_{n,m\in \mathbb N}$ a finite union of independent sets? (one can take any two primes $p,q\neq 2$.

Moreover, is this set a Sidon set? (Sidon sets are the following - the definition from: *Rudin, Walter*, Trigonometric series with gaps, J. Math. Mech. 9, 203-227 (1960); DOI: 10.1512/iumj.1960.9.59013, jstor. ZBL0091.05802, MR116177.)

(b) A set $E$ is said to be a

Sidon setif there is a finite constant $B$ such that $$\sum_{-\infty}^\infty |\hat{f}(n)| \le B \|f\|_\infty$$ for every $E$-polynomial $f$.

where $\hat{f}(n)$ is the $n$'th Fourier coefficient of $f$.

Thanks.