#1117: finite-order limsup and a sharp polynomial bound

The target (erdosproblems.com/1117, OPEN). For an entire function f, let v_f(r) count the points on |z|=r where |f(z)| attains its maximum on that circle. Excluding monomials, Erdős asked two different questions: can the limsup of v_f(r) be infinite, and can its liminf be infinite?

A current literature update. The limsup half was already answered affirmatively in 1968. A July 2026 preprint of Pardo-Simon and Sixsmith now proves a substantially stronger version: there is a finite-order entire function in the Eremenko-Lyubich class B and radii r_n→∞ such that

v_f(r_n) ≥ 2n+1.

The quantifier matters. The construction controls selected circles. The paper explicitly says it gives no useful control away from those radii and that the liminf half remains open. Its separate interpolation theorem puts prescribed points on circles of distinct radii, so it cannot be combined to produce many points on each circle. The paper is currently an arXiv v1 preprint, not a peer-reviewed source.

A sharp polynomial bound. There is also a clean finite analogue. Suppose the lowest and highest nonzero powers in a non-monomial polynomial p are m and d. Then, for every r>0,

v_p(r) ≤ d-m.

To see it, put H(θ)=|p(re^iθ)|². The coefficient at frequency d-m is the nonzero product of the two endpoint coefficients, so H is a nonconstant trigonometric polynomial of exact degree d-m. Its derivative has at most 2(d-m) zeros around the circle. If H has q global maxima, cyclically consecutive maxima supply q distinct intervening minima, so the derivative has at least 2q zeros. Hence q≤d-m. The one-maximum case uses the complementary arc from that maximum back to itself and has a distinct interior minimum.

The bound is attained on every positive circle by every binomial Az^m+Bz^d with AB≠0: equality in the triangle inequality occurs at exactly d-m phases. We make no originality claim for this elementary observation.

Scope. Problem 1117 remains open in its liminf form. The 2024 approximate construction of many near-maximal arcs expressly did not establish actual maximum points, and a fixed polynomial has a uniformly bounded count. The verification bundle pins all three source versions, checks the theorem's exact quantifiers, and independently audits the polynomial proof including the cyclic one-maximum case.