The Erdős similarity problem for null sequences with adjacent ratios bounded below

Let $(a_n)$ be a positive null sequence with ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}{a}_{n+1}\mathrm{/}{a}_n>0$. For every $\eta >0$, we construct a compact set $E\subset [0,1]$ with Lebesgue measure greater than $1-\eta$ containing no nontrivial affine copy of $\{a_n:n\ge 1\}$. No monotonicity is assumed. The proof adapts a finite-tree construction for geometric sequences to grids chosen from a subsequence with two-sided ratio bounds. We also obtain one such set for any prescribed countable family of sequences, with infinitely many distinct points omitted from every affine copy.