Embedding binary sequences into Bernoulli site percolation on Z3

We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on Zd with parameter p. In 1995, I. Benjamini and H. Kesten proved that, for d⩾10 and p=1/2, all sequences can be embedded, almost surely. They conjectured that the same should hold for d⩾3. We consider d⩾3 and p∈(pc(d),1−pc(d)), where pc(d)<1/2 is the critical threshold for site percolation on Zd. We show that there exists an integer M=M(p), such that, a.s., every binary sequence, for which every run of consecutive 0s or 1s contains at least M digits, can be embedded.