Williams's one-tape question: simulating one-tape time $T$ in $O(T^{1/2-\delta})$ space

Theorem 1.1: for a fixed deterministic machine with one writable tape and head (and finitely many read-only input heads), the finite-control and halting outcome by time $T$ can be computed in $O(T^{2/5}\log^C(T+2))$ work-space bits given the cap $T$; without a supplied cap, a machine halting at time $t$ is simulated in $O((t+2)^{2/5}\,\mathrm{polylog})$ space. Hence $O(T^{1/2-\delta})$ for every $\delta<1/10$. Simulation time is unrestricted, the final tape is not materialized, no lower bound or optimality is claimed, and for several writable tapes only $O(\sqrt T\,\mathrm{polylog})$ is given.

Theorem 1.1: for a fixed deterministic machine with one writable tape and head (and finitely many read-only input heads), the finite-control and halting outcome by time $T$ can be computed in $O(T^{2/5}\log^C(T+2))$ work-space bits given the cap $T$; without a supplied cap, a machine halting at time $t$ is simulated in $O((t+2)^{2/5}\,\mathrm{polylog})$ space. Hence $O(T^{1/2-\delta})$ for every $\delta<1/10$. Simulation time is unrestricted, the final tape is not materialized, no lower bound or optimality is claimed, and for several writable tapes only $O(\sqrt T\,\mathrm{polylog})$ is given.