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Solution of the Littlewood-Offord Problem in High Dimensions

P. Frankl and Z. Furedi
Annals of Mathematics
Second Series, Vol. 128, No. 2 (Sep., 1988), pp. 259-270
Published by: Annals of Mathematics
DOI: 10.2307/1971442
Stable URL: http://www.jstor.org/stable/1971442
Page Count: 12
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Solution of the Littlewood-Offord Problem in High Dimensions
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Abstract

Consider the 2n partial sums of arbitrary n vectors of length at least one in d-dimensional Euclidean space. It is shown that as n goes to infinity no closed ball of diameter Δ contains more than $(\lfloor\Delta\rfloor) + 1 + o(1)) ({n \atop \lfloor n/2 \rfloor})$ out of these sums and this is best possible. For $\Delta - \lfloor\Delta\rfloor$ small an exact formula is given.

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