Littlewood-Offord inequality ($\#\{ϵ:\Vert {\sum }_i{ϵ}_i{x}_i-a\Vert <1\}\le \left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$)

For vectors of norm at least one, the number of signed sums in an open unit ball is at most the central binomial coefficient. In one dimension this follows from the antichain bound. In Euclidean spaces a separated block decomposition for vector subset sums supplies the same bound without assuming all vectors have positive projection in one common direction. Signed sums are counted with multiplicity, by their sign choices.

For real coefficients of absolute value at least one, an open interval of length four contains at most the sum of the two largest binomial coefficients of signed sums. The corresponding subset family has no three-element inclusion chain. Its Lubell mass is therefore at most two, giving the bound by allocating mass to the two largest levels. Unit coefficients attain it at a center midway between the two chosen sum levels.

For Euclidean vectors $x_i$ of length at least one, their labeled subset sums can be partitioned into pairwise distance-at-least-one blocks, with $\left(\genfrac{}{}{0ex}{}{n}{r}\right)-\left(\genfrac{}{}{0ex}{}{n}{r-1}\right)$ blocks of size $n-2r+1$. To extend a separated block by a vector $x$, choose its member $y$ minimizing inner product with $x$. The translated block together with $y$ remains separated, since $(z+x-y)\cdot x\ge \Vert x{\Vert }^2$. The untransformed block with $y$ removed remains separated too. Block sizes change from $\ell$ to $\ell +1,\ell -1$, the same recurrence as a symmetric chain decomposition of a Boolean lattice. There are exactly $\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$ nonempty blocks.