One and two dimensional Cantor-Lebesgue type theorems

Let phi(n) be any function which grows more slowly than exponentially in n, i.e., [GRAPHICS] There is a double trigonometric series whose coefficients grow like phi(n), and which is everywhere convergent in the square, restricted rectangular, and one-way iterative senses. Given any preassigned rate, there is a one dimensional trigonometric series whose coefficients grow at that rate, but which has an everywhere convergent partial sum subsequence. There is a one dimensional trigonometric series whose coefficients grow like phi(n), and which has the everywhere convergent partial sum subsequence S-2j. For any p > 1, there is a one dimensional trigonometric series whose coefficients grow like phi(n (p-1/p)), and which has the everywhere convergent partial sum subsequence S-[jp]. All these examples exhibit, in a sense, the worst possible behavior. If m(j) is increasing and has arbitrarily large gaps, there is a one dimensional trigonometric series with unbounded coefficients which has the everywhere convergent partial sum subsequence S-mj.