Proof of a conditional theorem of Littlewood on the distribution of values of entire functions

It is proved that for any entire function $f$ of finite nonzero order there is a set $S$ in the plane with density zero and such that for any $a\in\mathbf C$ almost all the roots of the equation $f(z)=a$ belong to $S$. This assertion was deduced by Littlewood from an unproved conjecture about an estimate of the spherical derivative of a polynomial. This conjecture is proved here in a weakened form.
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