mjhjmtu

Can the ratio of $q$-smooth numbers ever approach one? Asked more rigorously, can we ever have
$$1in overlinea/b; :; a,b ;text are qtext-smooth=overlineprod_p_ile qp_i^alpha_i; :; (a_1, dots, a_pi(q))inmathbbZ^pi(q);?$$
If $q=2$, the answer is obviously no because
$$1notin overline2^alpha; :; alphainmathbbZ=0cup 2^alpha; :; alphainmathbbZ$$
however beyond this the problem is not easy. The reason I would like to know this is because this would imply that
$$G(q)=inf a/b; :; a>b,; a,b ; qtext-smooth>1.$$
I suspect that we may be able to say even further that
$$overlineprod_p_ile qp_i^alpha_i; :; (a_1, dots, a_pi(q))inmathbbZ^pi(q)=0cup prod_p_ile qp_i^alpha_i; :; (a_1, dots, a_pi(q))inmathbbZ^pi(q)$$
which would clearly imply the result.

Yes, if $qge 3$.

This is because $log_2(3)$ is irrational, so the fractional parts of $log_2(3^n)$ lie densely in $[0,1)$. In particular they can become arbitrarily small.

Yes. We can just consider fractions of the form $frac 3^a2^b$. The base $2$ log of this is $alog_2(3)-b$. Because $log_2(3)$ is irrational, we can choose $a,b$ to make this difference arbitrarily close to $0$, which means the ratio can be arbitrarily close to $1$.