 
Summary: A survey of multidimensional generalizations of Cantor's
uniqueness theorem for trigonometric series
J. M. Ash
Abstract. Georg Cantor's pointwise uniqueness theorem for one dimensional
trigonometric series says that if, for each x in [0; 2 ),
P
cneinx = 0, then all
cn = 0. In dimension d, d 2, we begin by assuming that for each x in
[0; 2 )d,
P
cneinx = 0 where n = (n1; : : : ; nd) and nx = n1x1 + + ndxd. It
is quite natural to group together all terms whose indices di¤er only by signs.
But here there are still several di¤erent natural interpretations of the in...nite
multiple sum, and, correspondingly, several di¤erent potential generalizations
of Cantor's Theorem. For example, in two dimensions, two natural methods
of convergence are circular convergence and square convergence. In the former
case, the generalization is true, and this has been known since 1971. In the
latter case, this is still an open question. In this historical survey, I will dis
cuss these two cases as well as the cases of iterated convergence, unrestricted
rectangular convergence, restricted rectangular convergence, and simplex con
