Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good lower bound for $$ \min_{1 \leq \ell_1 < \ell_2 \leq N} ~ \frac{n_{\ell_2}}{n_{\ell_1}}. $$

Remark 1: Since by assumption $n_{\ell_2} > n_{\ell_1}$ for $\ell_2 > \ell_1$, a trivial lower bound is 1.

Remark 2: By the prime number theorem, the largest number $n_N$ is of size roughly $e^{c k \log k}$ for some $c$. Thus an obvious lower bound is something like $$ \frac{e^{c k \log k}}{e^{c k \log k} -1} \approx 1 + \frac{1}{e^{c k \log k}}. $$ The questions is if an essential improvement of this simple lower bound is possible. In particular, it would be nice to get a lower bound of size roughly $$ 1 + \frac{1}{e^{c k}}. $$

(All constants $c$ in these statements are generic positive numbers, their specific values are not of interest for me. Also, I don't care about small values of $k$ or $N$, but about an asymptotic result.)

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good lower bound for $$ \min_{1 \leq \ell_1 < \ell_2 \leq N} ~ \frac{n_{\ell_2}}{n_{\ell_1}}. $$

Remark 1: Since by assumption $n_{\ell_2} > n_{\ell_1}$ for $\ell_2 > \ell_1$, a trivial lower bound is 1.

Remark 2: By the prime number theorem, the largest number $n_N$ is of size roughly $e^{c k \log k}$ for some $c$. Thus an obvious lower bound is something like $$ \frac{e^{c k \log k}}{e^{c k \log k} -1} \approx 1 + \frac{1}{e^{c k \log k}}. $$ The questions is if an essential improvement of this simple lower bound is possible. In particular, it would be nice to get a lower bound of size roughly $$ 1 + \frac{1}{e^{c k}}. $$

(All constants $c$ in these statements are generic positive numbers, their specific values are not of interest for me. Also, I don't care about small values of $k$ or $N$, but about an asymptotic result.)

Edit: It would already be very helpful to know that 1% (or any other fixed percentage) of all possible quotients of consecutive numbers $n_{\ell+1}/n_\ell$ satisfy the desired lower bound.

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good lower bound for $$ \min_{1 \leq k < \ell \leq N} ~ \frac{n_\ell}{n_k}. $$

Remark 1: Since by assumption $n_\ell > n_k$ for $\ell > k$, a trivial lower bound is 1.

Remark 2: By the prime number theorem, the largest number $n_N$ is of size roughly $e^{c k \log k}$ for some $c$. Thus an obvious lower bound is something like $$ \frac{e^{c k \log k}}{e^{c k \log k} -1} \approx 1 + \frac{1}{e^{c k \log k}}. $$ The questions is if an essential improvement of this simple lower bound is possible. In particular, it would be nice to get a lower bound of size roughly $$ 1 + \frac{1}{e^{c k}}. $$

(All constants $c$ in these statements are generic positive numbers, their specific values are not of interest for me. Also, I don't care about small values of $k$ or $N$, but about an asymptotic result.)

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good lower bound for $$ \min_{1 \leq \ell_1 < \ell_2 \leq N} ~ \frac{n_{\ell_2}}{n_{\ell_1}}. $$

Remark 1: Since by assumption $n_{\ell_2} > n_{\ell_1}$ for $\ell_2 > \ell_1$, a trivial lower bound is 1.

Remark 2: By the prime number theorem, the largest number $n_N$ is of size roughly $e^{c k \log k}$ for some $c$. Thus an obvious lower bound is something like $$ \frac{e^{c k \log k}}{e^{c k \log k} -1} \approx 1 + \frac{1}{e^{c k \log k}}. $$ The questions is if an essential improvement of this simple lower bound is possible. In particular, it would be nice to get a lower bound of size roughly $$ 1 + \frac{1}{e^{c k}}. $$

(All constants $c$ in these statements are generic positive numbers, their specific values are not of interest for me. Also, I don't care about small values of $k$ or $N$, but about an asymptotic result.)