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Claim. Beck, Irregularities of distribution. I, Acta Math. 159 (1987), 1–49, proves two bounds on the disc discrepancy of Problem 989. Theorem 2A (pp. 3–4) is stated for an arbitrary infinite discrete set $A\subseteq\mathbb{R}^2$ and an arbitrary convex body $K$ of inradius at least $1$: some similar copy of $K$, contracted by a factor $a_0\in(0,1]$, holds a number of points of $A$ that differs from its area by more than a constant multiple of $\sigma(\partial K)^{1/2}$, where $\sigma(\partial K)$ is the length of the boundary; the Remark closing section 4 (p. 49) adds that $a_0>c_{32}$, a positive constant depending only on the dimension (Beck writes $c_{32}(K)$, his $K$ being the dimension). For a disc of radius $r\ge1$ this gives a disc of radius $R\in[cr,r]$ with $\lvert\lvert A\cap C\rvert-\pi R^2\rvert\gg r^{1/2}$. In the notation of the problem, with $F(r)=\max_{R\le r}f(R)$, the running maximum that Erdős offers as an alternative in Part I of his 1964 problem paper (the 1964 card),

$$F(r)\gg r^{1/2}$$

for every infinite $A$, and in particular $f(r)$ is unbounded for every $A$. The upper bound (pp. 4–5) is a construction: for each $r$ there is a periodic set $A_r$: one sample of a jittered lattice in the box $[-M,M)^2$, with $M=[\operatorname{diam}K]+1$, extended periodically modulo that box (period $2M$), for which every disc of radius at most $r$ has error $O((r\log r)^{1/2})$; the set depends on $r$. So for each $r$ the smallest value of $F(r)$ over all infinite $A$ lies between $c\,r^{1/2}$ and $C\,(r\log r)^{1/2}$. The paper does not prove $f(r)\gg r^{1/2}$ at each fixed radius for every $A$, and it does not give one set $A$ with $f(r)\ll(r\log r)^{1/2}$ for all $r$; the site's remark states both forms without the two qualifications.

What is answered. The first question is answered yes: $f(r)$ is unbounded for every $A$. The second question is answered for $F(r)$ up to a factor $(\log r)^{1/2}$, between $r^{1/2}$ and $(r\log r)^{1/2}$; the growth of $f$ at a fixed radius is not determined by the paper. The claim answers the second question in the reading that the problem page's Formulation takes from Erdős's 1964 source. The claim value is answered: the problem asks how fast the discrepancy grows, a question whose answer is a growth rate rather than a yes or a no, and the two bounds are that answer for $F$ up to the logarithmic gap.

Depends on. Nothing in this wiki; the result rests on the cited paper alone.

Formalization. One public Lean development, not built or audited here, so no formalized evidence is listed. Collin Yuanjie Ren's package JSP-000822 (README of 2026-09-16, pinned above), which the community database's note for the problem cites and records as AI-assisted without naming the system, declares itself a formalization of Beck's absolute running-radius disc bounds and their unboundedness consequence; its root theorem proves that for every $R\ge R_0$ and every infinite locally finite $S$ some closed disc of radius $r\in(0,R]$ has discrepancy at least $c\sqrt R$, that for every $R\ge R_0$ there is one such $S$ for which every closed disc of radius at most $R$ has discrepancy at most $1000\sqrt{R\log R}$ (with $R_0=4$), and that no set has bounded disc discrepancy. Its README states that the lower bound chooses a radius up to $R$ and that the upper set may depend on $R$, and reports the axioms propext, Classical.choice and Quot.sound.

Dating. The paper prints "Imprimé le 25 août 1987" on its signature pages (pp. 1, 17, 33 and 49); it was received on 31 January 1985 and revised on 10 March 1986 (p. 49). The page is dated by the printed date.

Acceptance. Refereed: J. Beck, Irregularities of distribution. I, Acta Math. 159 (1987), 1–49. Reviewed: the site's curator, Thomas Bloom, labels the problem SOLVED and credits this paper with the result in the site's remark; the community database records the problem as solved and unformalized. The statement above follows the paper: Theorem 2A on pp. 3–4, the construction on pp. 4–5 and the Remark on p. 49.