On the sequence of numbers of the form $ε₀ + ε₁q + ... + ε_nq^n$, $ε_i ∈ {0,1}$

Paul Erdős; István Joó; Vilmos Komornik

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Let V be an origin-symmetric convex body in $ℝ^{n}$ , n≥ 2, of Gaussian measure $γ_{n} (V) \geq 1 / 2$ . It is proved that for every choice $u_{1}, . . ., u_{n}$ of vectors in the Euclidean unit ball $B_{n}$ , there exist signs $ε_{j} \in - 1, 1$ with $ε_{1} u_{1} + . . . + ε_{n} u_{n} \in (c l o g n) V$ . The method used can be modified to give simple proofs of several related results of J. Spencer and E. D. Gluskin.