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Let $\mathcal{P}_s$ be the lattice of degrees of non-empty $\Pi_1^0$ subsetsof $2^\omega$ under Medvedev reducibility. Binns and Simpson proved that$FD(\omega)$, the free distributive lattice on countably many generators, islattice-embeddable below any non-zero element in $\mathcal{P}_s$. Cenzer andHinman proved that $\mathcal{P}_s$ is dense, by adapting the Sacks Preservationand Sacks Coding Strategies used in the proof of the density of the c.e.\Turing degrees. With a construction that is a modification of the one by Cenzerand Hinman, we improve on the result of Binns and Simpson by showing that forany $\mathcal{U} <_s \mathcal{V}$, we can lattice embed $FD(\omega)$ into$\mathcal{P}_s$ strictly between $deg_s(\mathcal{U})$ and $deg_s(\mathcal{V)}$.We also note that, in contrast to the infinite injury in the proof of the SacksDensity Theorem, in our proof all injury is finite, and that this is also truefor the proof of Cenzer and Hinman, if a straightforward simplification ismade.

Embedding $FD(\omega)$ into $\mathcal{P}_s$ densely