A Note on a Problem of Hilliker and Straus

Abstract

For a prime $p$ and a vector $\bar\alpha=(\alpha_1,\dots,\alpha_k)\in {\Bbb Z}_p^k$ let $f\left(\bar\alpha,p\right)$ be the largest $n$ such that in each set $A\subseteq{\Bbb Z}_{p}$ of $n$ elements one can find $x$ which has a unique representation in the form $x=\alpha_{1}a_1+\dots +\alpha_{k}a_k, a_i\in A$. Hilliker and Straus bounded $f\left(\bar\alpha,p\right)$ from below by an expression which contained the $L_1$-norm of $\bar\alpha$ and asked if there exists a positive constant $c\left(k\right)$ so that $f\left(\bar\alpha,p\right)>c\left(k\right)\log p$. 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 $A\subseteq {\Bbb Z}_{p}$ such that every element of $A+A$ has at least $K$ representations in the form $a+a'$, $a, a'\in A$.