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We derive new upper bounds for the densities of measurable sets in ${ℝ}^n$ which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions $2,\cdots ,24$. This gives new lower bounds for the measurable chromatic number in dimensions $3,\cdots ,24$. We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg,...