Roth criterion for rational Fredholm values ($l(1-\log p/\log q)>2$)

For $0<p<q$, truncating the Fredholm series at $n$ gives a rational with denominator at most $Q_n=q^{l^n}$ and positive error at most $2Q_n^{-l(1-\log p/\log q)}$ for large $n$. The displayed strict inequality first rules out a rational value by the rational separation bound, and then rules out an algebraic irrational value by Roth theorem. Reduced denominators are at most $Q_n$ and tend to infinity because the truncations are distinct. In particular $l\ge 3$ and $p<q^{\delta }$, $\delta <1/3$, imply transcendence.