Bounds for arrays of dots with distinct slopes or lengths
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Ann×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−Cn11/20<m<n+4n2/3 for alln andm>n+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
AMS subject classification code (1991)05 C
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