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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.

Claimant and date. The deduction is the commentary of erdosproblems.com's own problem page, unsigned, whose recommended citation names the erdosproblems.com curator, T. F. Bloom, as the page's author; the page is filed under that name. The commentary 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. Their paper does not state the union bound; its Theorem 1.1 is the source of the eventual bound the deduction uses. The erdosproblems.com 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. erdosproblems.com page's history was not read here; the erdosproblems.com page, last edited on 23 January 2026, carried the deduction then, so that date, the earliest this corpus attests for the deduction, dates this page. The deduction is written in full on the library's corollary page, which is Plasma AI's check of it. 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.

Standing. Pending. Both inputs are refereed. The first is Theorem 1.1 of Chenxu Hao, Xinyi Li, Izumi Okada and Yushu Zheng in Probability Theory and Related Fields 195 (2026), 1765–1822, published online 12 November 2025 (doi:10.1007/s00440-025-01441-1; first posted as arXiv:2409.00995 on 2 September 2024). The second is 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 published only in the erdosproblems.com commentary, so refereed is not listed. The curator who labeled the problem proved wrote the deduction, so the label is the claimant's own and not independent acceptance, and no reviewed evidence comes from it. Plasma AI checked the deduction in full on the corollary page; acceptance on that check is pending until the project's reviewer signs off, so no evidence is listed. An independent Lean proof of the deduction is recorded on its own claim page.

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.