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Claim. For the walk law of [[problems/analysis/E1166/_index|Problem 1166]], discrete-time symmetric nearest-neighbor simple random walk on $\mathbb Z^2$ started at the origin, almost surely

$$\left|\bigcup_{k\le n}F(k)\right|=O((\log n)^2),$$

so the question has a positive answer with exponent $2$. Two refereed theorems combine to give it. Theorem 1.1 of Chenxu Hao, Xinyi Li, Izumi Okada and Yushu Zheng, Favorite sites for simple random walk in two and more dimensions, recorded as Theorem 1.1, proves that $\limsup_n|F(n)|=3$ almost surely, so $|F(n)|\le3$ for all large $n$; this is the content of Problem 1165. The planar estimate of Erdős and Taylor, recorded as the maximum-local-time upper bound, gives $\limsup_nT_n/(\log n)^2\le1/\pi$ almost surely, where $T_n=\max_xf_n(x)$. While the maximum local time stays at one level the favorite sets only grow, so once their size is at most three each level contributes at most three sites to the union; there are at most $T_n$ levels by time $n$, whence

$$\left|\bigcup_{k\le n}F(k)\right|\le C_\omega+3T_n$$

for a finite random constant $C_\omega$, and the limit superior of the left side over $(\log n)^2$ is at most $3/\pi$. That constant is an upper bound only; no sharp asymptotic for the union is asserted.

Attribution. The union bound is not a numbered statement of the paper. The site's page for Problem 1166 states the deduction itself: it derives the bound from the eventual bound $|F(n)|\le3$, for which it points to Problem 1165, and from the Erdős–Taylor estimate, and it does not name Hao, Li, Okada and Zheng. The site's page for Problem 1165 credits that eventual bound (probability $0$ for four or more favorites) to Tóth's 2001 paper, which concerns the walk on $\mathbb Z$, a misattribution that the E1165 claim page records; in the plane Theorem 1.1 supplies it. This page files the result under the paper's authors because their theorem is the input that was missing; the 1960 estimate is classical. The deduction is written in full on the library's corollary page, which is this corpus's own writing and awards no evidence. The question is Problem 6.78 of the 1999 booklet Some of Paul's favorite problems, attributed there to Erdős and Révész, whose union starts at $k=1$; including $F(0)$ changes the count by at most one.

Acceptance. Both inputs are refereed: Theorem 1.1 in Probability Theory and Related Fields 195 (2026), 1765–1822, published online 12 November 2025, and P. Erdős and S. J. Taylor, Some problems concerning the structure of random walk paths, Acta Math. Acad. Sci. Hungar. 11 (1960), 137–162. The deduction itself is unpublished, so refereed is not listed. The site's curator, Thomas F. Bloom, states the deduction and marks the problem proved, but credits the eventual bound to Tóth rather than to this claimant, so no reviewed evidence is listed and the claim stays claimed. An independent Lean proof of the deduction is recorded on its own claim page. The page is dated by the first arXiv posting of the paper, 2 September 2024.

Depends on. The accepted claim Hao, Li, Okada and Zheng's three favorite sites of Problem 1165 supplies the eventual bound; the written deduction and its inputs are the library pages linked above.