A nonuniform Littlewood–Offord inequality for all norms

Let vi be vectors in Rd and {εi} independent Rademacher random variables. Then the Littlewood–Offord problem entails finding best upper bound for supx∈RdP(∑εivi=x). Generalizing uniform bounds of Littlewood–Offord, Erdős Kleitman, a recent result Dzindzalieta Juškevičius provides non-uniform that is optimal its dependence on ‖x‖2. In this short note, we provide simple alternative proof their result. Furthermore, our demonstrates applies to any norm Rd, not just ℓ2 norm. This resolves conjecture Juškevičius.