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With the help of these estimates, we recover results about the size of $A \\cdot A+...+A \\cdot A$, where $A$ is a subset of the real line of a given Hausdorff dimension, $A+A=\\{a+a': a,a' \\in A \\}$ and $A \\cdot A=\\{a \\cdot a': a,a' \\in A\\}$. We also use projection results and inductive arguments to show that if a Hausdorff dimension of a subset of ${\\Bbb R}^d$ is sufficiently large, then the ${k+1 \\choose 2}$-dimensional Lebesgue measure of the set of $k$-simplexes determined by this set is positive. 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