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Is there an entire non-zero function such that, for any infinite sequence , the set
is everywhere dense?
Is there an entire transcendental function such that, for any infinite sequence , the set
is everywhere dense?
Source: erdosproblems.com/906
A full solution has been claimed but not yet accepted. The statement is true.
The site labels the problem OPEN (page last edited 1 October 2025), a label that describes the corrected Statement. Four pending full claims, each with declared AI assistance, construct such a transcendental function: Almeida's planar Gaussian series and Chojecki's Gaussian series, both posted on the site's discussion thread on 25 April 2026, and, on the site's proof-claims tab, Hou's random Fock series of 21 July 2026 and He's sparse Fock series of 17 September 2026, the last of which does not claim to be the first solution. A 1973 theorem of Boas and Reddy, as printed, excludes three of the four constructions, and the conflict is unresolved. The site has accepted none of them, this corpus has built neither of the two Lean developments, and the derived standing is claimed.
The site's wording is trivially true: every non-zero polynomial, the constant among them, is an entire non-zero function whose derivatives of order above its degree vanish identically, so for every sequence the set contains the whole plane. The failure covers the whole class of polynomials, so it is a failure of setting, not of range. Tang pointed it out, as the site's commentary records. The change replaces "non-zero" by "transcendental"; a transcendental entire function is non-zero, and nothing else changes. The evidence is the site's own commentary, which records the polynomial failure and gives the intended form, transcendental, while the site keeps the label OPEN, which only that form fits; the formal-conjectures statement file, which counts with the site, states the same form. The defect is already in the poser's text: Erdős's question (i) in [Er82e, p. 72, §IV.1] asks for an entire function with the roots of the everywhere dense, with no condition that excludes polynomials, and the site's "non-zero" excludes only the zero function. The correction does not rest on Erdős's Hungarian paper (Some remarks on a paper of Kővári, Mat. Lapok 7 (1956), 214--217), where [Er82e] says the question was raised, or on [BaSc72]. The form follows from the site's commentary and label, not from the results that settle it. The polynomial observation answers the site's wording only and counts for nothing; no claim page records it, since no one presented it as settling the problem. The page's standing judges the corrected Statement.