= Fredholm lacunary series
{c}
{title2=$f_l(z)=\sum_{n\geq0}z^{l^n},\qquad l\geq2$}
= Fredholm series
{c}
{synonym}
This lacunary series is analytic for $|z|<1$ and satisfies $f_l(z^l)=f_l(z)-z$. At roots of unity of $l$-power order, its radial tail diverges, giving a dense set of singularities on the unit circle and excluding algebraicity as a function. The <Mahler method> proves transcendence of its values at every algebraic $z$ with $0<|z|<1$. At $z=1/2$, reduced truncations have denominators $2^{l^m}$ and error less than twice their $-l$th power; for $l\geq3$, the <Roth theorem> gives a shorter proof.
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