  The Erd\H{o}s similarity conjecture asserted that an infinite set of real
numbers cannot be affinely embedded into every measurable set of positive
Lebesgue measure. The problem is still open, in particular for all fast
decaying sequences. In this paper, we relax the problem to the bi-Lipschitz
embedding and obtain some sharp criteria about the bi-Lipschitz Erd\H{o}s
similarity problem for strictly decreasing sequences.
