A Log-Star Comparison Between Expectation Threshold and Fractional Expectation Threshold

We proved a log-star comparison between the expectation threshold $q(\mathcal{F})$ and the fractional expectation threshold $q_f(\mathcal{F})$ for any nontrivial increasing family $\mathcal{F}$ on a finite ground set $V$ of size $\mathrm{\mid }V\mathrm{\mid }=N$. Specifically, we show that $$q_f(\mathcal{F})\le 64{\log }_2^{\ast }(N+2)q(\mathcal{F}),$$ where we define ${\log }_2^{\ast }x$ to be the least integer $k\ge 0$ such that applying ${\log }_2$ repeatedly $k$ times gives a number at most $1$.