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Claim. For real $x_1,\ldots,x_n$ with $|x_i|\ge1$, at most $\binom n{\lfloor n/2\rfloor}$ sign choices put $\sum_i\epsilon_ix_i$ in any open interval of length two. This is Theorem 1 of P. Erdős, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc. 51 (1945), no. 12, 898–902 (result page). Whatever its center, an open disc of radius one meets the real line in an open interval of length at most two, or not at all. So the bound is Problem 498 for real coefficients. The same paper gives only the weaker bound $O(2^n/\sqrt n)$ for complex coefficients, which settles no instance of the problem.

Covers. Problem 498 when every $z_i$ is real, with $D$ read as an open disc.

Acceptance. A refereed journal publication, the refereed evidence; the page heads give the issue as December 1945 and no day, so the page is dated to the first day of that month. The site's curator, Thomas Bloom, labels the problem proved and credits Erdős with the real case, the reviewed evidence.

Depends on. Nothing beyond the cited paper.