The geodesic structures on self-similar fractals are interesting. One feature of geodesic structures is average distance on fractal. We investigate the Sierpinski-like carpet and obtain the average distance in terms of self-similar measure.

For self-similar set [Formula: see text] [Formula: see text] we study the arithmetic product of [Formula: see text] and obtain that when [Formula: see text] then [Formula: see text] To deal with the arithmetic product of self-similar sets with different ratios, we introduce a technique named comparable rectangle.