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$\times 2\times 3$ theorem
Furstenberg’s 
$\times 2\times 3$ theorem asserts that the double sequence 
$(2^{m}3^{n}\unicode[STIX]{x1D6FC})_{m,n\geq 1}$ is dense modulo one for every irrational 
$\unicode[STIX]{x1D6FC}$. The same holds with 
$2$ and 
$3$ replaced by any two multiplicatively independent integers. Here we obtain the same result for the sequences 
$((\begin{smallmatrix}m+n\\ d\end{smallmatrix})a^{m}b^{n}\unicode[STIX]{x1D6FC})_{m,n\geq 1}$ for any non-negative integer 
$d$ and irrational 
$\unicode[STIX]{x1D6FC}$, and for the sequence 
$(P(m)a^{m}b^{n})_{m,n\geq 1}$, where 
$P$ is any polynomial with at least one irrational coefficient. Similarly to Furstenberg’s theorem, both results are obtained by considering appropriate dynamical systems.
