Abstract.
Given any bijection f: Z r→f: Z s with s≥ r , easy volume comparisons show that there must be a universal constant K>0 (depending only on r and s ) and infinitely many pairs of points x,y∈ Z r such that || f(x)-f(y)|| > K|| x-y|| r/s . This puts a bound on how much contraction can be achieved for any such bijection. We show that, conversely, for any s≥ r there is a bijection f: Z r→Z s and a constant C>0 such that for all x,y∈ Z r we have || f(x)-f(y)|| <C|| x-y|| r/s . Phrased differently there is a bijection f: Z r→Z s which shrinks the distance between the images of any two points as much as possible, up to a constant factor. This generalizes a construction in fractal image processing and answers in the affirmative a question of Michael Freedman.
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Received May 15, 1996.
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Stong, R. Mapping Zr into Zs with Maximal Contraction . Discrete Comput Geom 20, 131–138 (1998). https://doi.org/10.1007/PL00009375
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DOI: https://doi.org/10.1007/PL00009375