Let ϑ be an irrational number and let { t } denote the fractional part of t . For each N let I 0 , I 1 , … , I N be the intervals resulting from the partition of [ 0 , 1 ] by the points { k 2 ϑ } , k = 1 , 2 , … , N . Let T ( N ) be the number of distinct lengths these intervals can assume. It is shown that T ( N ) → ∞ . This is in contrast to the case of the sequence { n ϑ } , where T ( N ) ≤ 3 .