Summary: A symmetric subset of the reals is one that remains invariant under some reflection \(x\mapsto c- x\). We consider, for any \(0< \varepsilon\leq 1\), the largest real number \(\Delta(\varepsilon)\) such that every subset of \([0,1]\) with measure greater than \(\varepsilon\) contains a symmetric subset with measure \(\Delta(\varepsilon)\). In this paper we establish upper and lower bounds for \(\Delta(\varepsilon)\) of the same order of magnitude: For example, we prove that \(\Delta(\varepsilon)= 2\varepsilon- 1\) for \({11\over 16}\leq\varepsilon\leq 1\) and that \(0.59\varepsilon^2< \Delta(\varepsilon)< 0.8\varepsilon^2\) for \(0< \varepsilon\leq{11\over 16}\).
This continuous problem is intimately connected with a corresponding discrete problem. A set \(S\) of integers is called a \(B^*[g]\) set if for any given \(m\) there are at most \(g\) ordered pairs \((s_1,s_2)\in S\times S\) with \(s_1+ s_2= m\); in the case \(g= 2\), these are better known as Sidon sets. Our lower bound on \(\Delta(\varepsilon)\) implies that every \(B^*[g]\) set contained in \(\{1,2,\dots, n\}\) has cardinality less than \(1.30036\sqrt{gn}\). This improves a result of Green for \(g\geq 30\). Conversely, we use a probabilistic construction of \(B^*[g]\) sets to establish an upper bound on \(\Delta(\varepsilon)\) for small \(\varepsilon\).