LAUSR

Abstract Given a sequence z ¯ = ( z 1 , z 2 , … ) where z k ∈ [ 0 , 1 ] d , the dispersion of z ¯ is defined by μ d ( z ¯ ) = inf n ≥ 1 inf m ≥ 1 n 1 / d ‖ z m − z m + n ‖ , where ‖ ⋅ ‖ is taken to be the L 1 metric ‖ ⋅ ‖ 1 on [ 0 , 1 ] d . This is a natural measure for how “spread out” the sequence z ¯ is. In this paper, we investigate α d : = sup z ¯ μ d ( z ¯ ) . Note that the requirement μ d ( z ¯ ) = α d > 0 is a very stringent condition on the distribution of the z i . For example, it implies that ‖ z m − z m + 1 ‖ ≥ α d for all m ≥ 1 . We show by construction that α d ≥ ( ( 2 d − 1 ( 2 d − 1 ) ) 1 / d ( 1 + ∑ k ≥ 1 1 F 2 k ) ) − 1 > 0.098 where F n denotes the n t h Fibonacci number. We also introduce a combinatorial problem for d-tuples of permutations which yields bounds on α d . This work extends previous results of the authors where the value of α 1 is determined.