A concentration function estimate and intersective sets from matrices

We give several sufficient conditions on an infinite integer matrix (d _ij ) for the set R = {Σ _{ij∈α, i>j} d _ij : α ? ?, |α| < ∞} to be a density intersective set, including the cases d _nj = j ⁿ (1 + O(1/n ^1+ε )) and \(0 < d_{nj} = o(\sqrt {n/\log n} )\). For the latter, a concentration function estimate that is of independent interest is applied to sums of sequences of 2-valued random variables whose means may grow as \(\sqrt {n/\log n} \).