  For a fixed unit vector a=(a_1,a_2,...,a_n) in S^{n-1}, i.e. sum_{i=1}^n
a_i^2=1, we consider the 2^n sign vectors
epsilon=(epsilon_1,epsilon_2,...,epsilon_n) in {-1,1}^n and the corresponding
scalar products a.epsilon=sum_{i=1}^n a_i epsilon_i. Holtzman and Kleitman
formulated the following conjecture. It states that among the 2^n sums of the
form sum +/- a_i there are not more with |sum_{i=1}^n +/- a_i|>1 than there are
with |sum_{i=1}^n +/- a_i| <= 1. The result is of interest in itself, but has
also an appealing reformulation in probability theory and in geometry. In this
paper we will solve an extension of this problem in the uniform case where all
the a's are equal. More precisely, for S_n being a sum of n independent
Rademacher random variables, we will give, for several values of xi, precise
lower bounds for the probabilities P_n:=P{-xi sqrt{n} <= S_n <= xi sqrt{n}}.
There is an obvious relationship with the binomial distribution with parameters
n and p=1/2.
  The obtained lower bounds are sharp and much better than for instance the
bound that can be obtained from application of the Chebishev inequality. In
case xi=1 Van Zuijlen solved this problem. We remark that our bound will have
nice applications in probability theory and especially in random walk theory.
