Proof of Tomaszewski's conjecture on randomly signed sums
نویسندگان
چکیده
We prove the following conjecture, due to Tomaszewski (1986): Let X=∑i=1naixi, where ∑iai2=1 and each xi is a uniformly random sign. Then Pr[|X|≤1]≥1/2. Our main novel tools are local concentration inequalities an improved Berry-Esseen inequality for Rademacher sums.
منابع مشابه
Tomaszewski's Problem on Randomly Signed Sums: Breaking the 3/8 Barrier
Let v1, v2, . . . , vn be real numbers whose squares add up to 1. Consider the 2 n signed sums of the form S = ∑ ±vi. Holzman and Kleitman (1992) proved that at least 38 of these sums satisfy |S| 6 1. This 3 8 bound seems to be the best their method can achieve. Using a different method, we improve the bound to 13 32 , thus breaking the 38 barrier.
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ژورنال
عنوان ژورنال: Advances in Mathematics
سال: 2022
ISSN: ['1857-8365', '1857-8438']
DOI: https://doi.org/10.1016/j.aim.2022.108558