Sums with curves (Q1071883)

Let \(C\subseteq {\mathbb{R}}^ 2\) be a fixed compact set and let \(E\subseteq {\mathbb{R}}^ 2\) be a Borel set. We form the sum \(E+C=\{e+c: e\in E,\quad c\in C\},\) which is a measurable set, and inquire as to the behavior of the two-dimensional Lebesgue measure \(m(E+C)\) as m(E)\(\to 0\). If C is ''thin'' then we expect that \(m(E+C)\) can be made small by choosing a suitable E. It is a consequence of the theorem below that if C is a curve (continuous image of [0,1]) which is not a line segment, then \(m(E+C)\) cannot, however, be small when compared to m(E). Theorem. If C is a curve and E is a Borel set, then \[ m(E+C)\geq [m(C-C)\cdot m(E)]^{1/2}/13\sqrt{2}. \] (Later results are in the author's paper ''The size of sums of sets'', to appear in Stud. Math..)