Some new results on the Littlewood-Offord problem

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Abstract

We consider the 2n sums of the form Σϵiai with the ai's vectors, | ai | ⩾ 1, and ϵi = 0, 1 for each i. We raise a number of questions about their distribution.

We show that if the ai lie in two dimensions, then at most n(n2)) sums can lie within a circle of diameter √3, and if n is even at most the sum of the three largest binomial coefficients can lie in a circle of diameter √5. These are best results under the indicated conditions.

If two a's are more than 60° but less than 120° apart in direction, then the bound (n[n2]) on sums lying within a unit diameter sphere is improved to (n+1[n2]) − 2(n−1[(n−12)]).

The method of Katona and Kleitman is shown to lead to a significant improvement on their two dimensional result.

Finally, Lubell-type relations for sums lying in a unit diameter sphere are examined.

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