A note on a problem of Hilliker and Straus (Q1010596)

Summary: For a prime \(p\) and a vector \(\bar\alpha=(\alpha_1,\dots,\alpha_k)\in {\mathbb Z}_p^k\) let \(f(\bar{\alpha},p)\) be the largest \(n\) such that in each set \(A\subseteq{\mathbb Z}_{p}\) of \(n\) elements one can find \(x\) which has a unique representation in the form \(x=\alpha_{1}a_1+\dots +\alpha_{k}a_k,a_i\in A\). \textit{D. L. Hilliker} and \textit{E. G. Straus} [J. Number Theory 24, 1--6 (1986; Zbl 0589.10059)] bounded \(f(\bar{\alpha},p)\) from below by an expression which contains the \(L_1\)-norm of \(\bar{\alpha}\) and asked if there exists a positive constant \(c(k)\) so that \(f(\bar{\alpha},p)>c(k)\log p\). In this note we answer their question in the affirmative and show that, for large \(k\), one can take \(c(k)=O(1/k\log (2k)) \). We also give a lower bound for the size of a set \(A\subseteq {\mathbb Z}_{p}\) such that every element of \(A+A\) has at least \(K\) representations in the form \(a+a^{\prime}, a, a^{\prime}\in A\).