Bounds for arrays of dots with distinct slopes or lengths

AbstractAnn×m sonar sequence is a subset of then×m grid with exactly one point in each column, such that the $$\mathop 2\limits^m $$ vectors determined by them are all distinct. We show that for fixedn the maximalm for which a sonar sequence exists satisfiesn−Cn11/20n+c logn log logn for infinitely manyn.Another problem concerns the maximal numberD of points that can be selected from then×m grid so that all the $$\mathop 2\limits^D $$ vectors have slopes. We proven1/2≪D≪n4/5