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Thickness theorems with partial derivatives

. In this paper, we prove some new thickness theorems with partial derivatives. We give some applications. First, we prove under some checkable conditions that the continuous image of arbitrary self-similar sets with positive similarity ratios is a closed interval, a ﬁnite union of closed intervals or containing interior. Second, we prove an analogous Erd˝os-Straus conjecture on the middle-third Cantor set.