Under the continuum hypothesis W. Sierpinski [7] proved that a set E which possesses 'the property C' is of measure zero with respect to any Hausdorff measure but E-E=R^1. In his proof we can see that a difference set A-B is closely related to the orthogonal projection of the product set A×B in the xy-plane to the line y= -x. In [8] D. J. Ward defined an n-difference set D^r(E) of a non empty set E⊂R^1 and showed that dim D (E)≦min {nα, n-1} under the conditions that the set E is an α-set and it has positive lower density with respect to the α-dimensional Hausdorff measure at every point in it.
In this iemark we shall estimate the lower and upper bounds of fractional dimensions of difference sets and show that the upper bound is sharp.
In §1, following [4] we shall define a perfect set of translation and under some condition we shall evaluate the Hausdorff measure of it in §2. In §3 we shall discuss the fractional dimensions of difference sets.