Lipschitz embeddings of random sequences

We develop a new multi-scale framework flexible enough to solve a number of problems involving embedding random sequences into random sequences. Grimmett et al. (Random Str Algorithm 37(1):85–99, 2010) asked whether there exists an increasing M-Lipschitz embedding from one i.i.d. Bernoulli sequence into an independent copy with positive probability. We give a positive answer for large enough M. A closely related problem is to show that two independent Poisson processes on ℝ are roughly isometric (or quasi-isometric). Our approach also applies in this case answering a conjecture of Szegedy and of Peled (Ann Appl Probab 20:462–494, 2010). Our theorem also gives a new proof to Winkler’s compatible sequences problem. Our approach does not explicitly depend on the particular geometry of the problems and we believe it will be applicable to a range of multi-scale and random embedding problems.