In this paper, we reduce the logarithmic Sarnak conjecture to the $\{0,1\}$-symbolic systems with polynomial mean complexity. By showing that the logarithmic Sarnak conjecture holds for any topologically dynamical system with sublinear complexity, we provide a variant of the $1$-Fourier uniformity conjecture, where the frequencies are restricted to any subset of $[0,1]$ with packing dimension less than one.

We introduce a full NT-step infeasible interior-point algorithm for semidefinite optimization based on a self-regular function to provide the feasibility step and to measure proximity to the central path. The result of polynomial complexity coincides with the best known iteration bound for infeasible interior-point methods.

In this paper, using the framework of self-regularity, we propose a hybrid adaptive algorithm for the linear optimization problem. If the current iterates are far from a central path, the algorithm employs a self-regular search direction, otherwise the classical Newton search direction is employed. This feature of the algorithm allows us to prove a worst case iteration bound. Our result matches the best iteration bound obtained by the pure self-regular approach and improves on the worst case iteration bound of the classical algorithm.