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{\bf Miros\l awa Ja\'{n}czak}
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{\bf A Note on a Problem of Hilliker and Straus}
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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 $a+a'$, $a, a'\in
A$.
\bye