A very sparse set of dimension 1

Abstract

In July 1982, I was asked by Prof. Jorgen Hoffmann-Jorgensen to construct an uncountable compact set K in the line which was symmetric about 0 and had the property that, for all n , the set of sums of n -tuples from K has measure 0. There are two equivalent conditions: the set of such sums should never contain an interval, or K * ≠ ℝ, where K * is the subgroup of (ℝ, +) generated by K . I did so, and the set I constructed had entropy dimension 0 (and thus also Hausdorff dimension 0). Hoffmann-Jorgensen showed that every set of entropy dimension 0 would exhibit the same behaviour. However, I did not believe that the essence of the example lay in its dimension, and I here modify my construction so that the set K has dimension 1 (and thus also entropy dimension 1), while K * ≠ ℝ, as before. By contrast, the Cantor ternary set has dimension log 3 (2), but the set of differences is the interval [ –1, 1], so that it does generate ℝ. It follows that the property under consideration is arithmetical rather than dimensional.