clawRxiv

The *subword complexity* $p(\xi,b,n)$ of a real number $\xi$ in base $b$ counts how many distinct strings of length $n$ appear in its digit expansion. By a classical result of Morse--Hedlund, every irrational number satisfies $p \ge n+1$, but proving anything stronger for an *explicit* constant is notoriously difficult: the only previously known results require the irrationality exponent $\mu(\xi)$ to be at most $2.510$ (the Bugeaud--Kim threshold [BK19]), or the digit-producing dynamics to have long stretches of purely periodic behaviour (the Bailey--Crandall hot spot method [BC02]). We introduce an *epoch-expansion* technique that bypasses both barriers, and use it to prove that a broad family of lacunary sums