Problem 498

Let $z_1,\ldots,z_n\in\mathbb{C}$ with $1\leq \lvert z_i\rvert$ for $1\leq i\leq n$ . Let $D$ be an arbitrary disc of radius $1$ . Is it true that the number of sums of the shape

\sum_{i=1}^n\epsilon_iz_i \textrm{ for }\epsilon_i\in \{-1,1\}

which lie in $D$ is at most $\binom{n}{\lfloor n/2\rfloor}$ ?

Source: erdosproblems.com/498

Combinatorics
Analysis

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SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN), the site's label. The site's Lean label is qualified in the formalization section below. The accepted claims are Kleitman's Theorem I with its open-disc consequence and Kleitman's Hilbert-space theorem, both refereed and credited by the site's curator, and Erdős's real case, an accepted partial claim.

Problem 498

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