Colbourn (1999) developed some strategy for nonadaptive group testing when the items are linearly ordered and the positives items form a consecutive subset of all items. We improve his strategy by introducing the concept of 2-consecutive positive detectable matrices (2CPD-matrix) requiring that all columns and bitwise OR-sum of each two consecutive columns are pairwise distinct. Such a matrix i...

We show that if \(A=\{a_1< a_2< \ldots < a_k\}\) is a set of real numbers such the differences consecutive elements are distinct, then for and finite \(B \subset \mathbb {R}\), $$ |A+B|\gg |A|^{1/2}|B|. $$ The bound tight up to constant.