Solved, the site's label: Beck [Be87] proved that for every infinite $A$ and every $r\ge1$ some disc of radius in $[cr,r]$ has discrepancy $\gg r^{1/2}$, so the running maximum $F(r)=\max_{R\le r}f(R)$ satisfies $F(r)\gg r^{1/2}$ and $f(r)$ is unbounded for every $A$, and that for each $r$ a periodic set keeps every disc of radius at most $r$ within $O((r\log r)^{1/2})$; the bounds concern $F(r)$ and an $r$-dependent construction, not $f$ at a fixed radius for one set, and the extremal growth of $F$ is known up to a factor $(\log r)^{1/2}$. The accepted claim is Beck 1987.