AbstractWe disprove a conjecture of Leo Moser by showing that (i) for every natural numbernand 0<α<2 there is a system ofnpoints on the unit sphereS2such that the number of pairs at distanceαfrom each other is at least const·nlog*n(where log* stands for the iterated logarithm function) (ii) for everynthere is a system ofnpoints onS2such that the number of pairs at distance√2 from each other is at least const -n4/3. We also construct a set ofnpoints in the plane in general position (no 3 on a line, no 4 on a circle) such that they determine fewer than const·nlog3/log2distinct distances, which settles a problem of Erdös.