Degrees and prime power order zeros of characters of symmetric and alternating groups

We show that the p$p$-part of the degree of an irreducible character of a symmetric group is completely determined by the set of vanishing elements of p$p$-power order. As a corollary, we deduce that the set of zeros of prime power order controls the degree of such a character. The same problem is analysed for alternating groups, where we show that when p=2$p=2$ these data can only be determined up to two possibilities. We prove analogous statements for the defect of the p$p$-block containing the character and for the p$p$-height of the character.