Boshernitzan’s condition, factor complexity, and an application

Boshernitzan gave a decay condition on the measure of cylinder sets that implies unique ergodicity for minimal subshifts. Interest in properties subshifts satisfying this has grown recently, due to connection with discrete Schrödinger operators, and particular interest is how restrictive is. While it zero topological entropy, our main theorem shows construct condition, whose factor complexity grows faster than any pre-assigned subexponential rate. As an application, via Damanik Lenz, we show there no subexponentially growing sequence which spectra all operators associated given have only finitely many gaps.