Tomaszewski's Problem on Randomly Signed sums, Revisited

Let $v_1,v_2,\ldots, v_n$ be real numbers whose squares add up to 1. Consider the $2^n$ signed sums of the form $S = \sum \pm v_i$. Boppana and Holzman (2017) proved that at least 13/32 of these sums satisfy $|S| \le 1$. Here we improve their bound to $0.427685$.

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Bibliographische Detailangaben
Veröffentlicht in:	The Electronic journal of combinatorics 2021-06, Vol.28 (2)
Hauptverfasser:	Boppana, Ravi B., Hendriks, Harrie, Van Zuijlen, Martien C.A.
Format:	Artikel
Sprache:	eng
Online-Zugang:	Volltext
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Beschreibung
Zusammenfassung:	Let $v_1,v_2,\ldots, v_n$ be real numbers whose squares add up to 1. Consider the $2^n$ signed sums of the form $S = \sum \pm v_i$. Boppana and Holzman (2017) proved that at least 13/32 of these sums satisfy $\|S\| \le 1$. Here we improve their bound to $0.427685$.
ISSN:	1077-8926 1077-8926
DOI:	10.37236/9497