Uniqueness of linear combinations \(\pmod p\) (Q1073833)

Let \(\bar\alpha=(\alpha_ 1,\dots,\alpha_ k)\) be a vector with nonzero integral components. To any set of residues \(S=\{a_ 1,\dots,a_ n\}\pmod p\) assign the set of residues \(L=[\alpha_ 1 a_{i_ 1}+\dots+\alpha_ k a_{i_ k} : a_{i_ j}\in S\}.\) The authors investigate the maximum number \(f(\bar\alpha;p)=n\) such that at least one element of \(L\) has a unique representation \(\pmod p\), obtaining results such as: if \(\alpha_ i=\alpha_ j\) or \(\alpha_ i=-\alpha_ j\) for some \(i\not\equiv j\), then \(f(\bar\alpha;p)<(2+\varepsilon)\log p/\log 3.\) It is conjectured that there is a constant \(c(\bar\alpha)\) such that \(f(\bar\alpha;p)<c(\bar\alpha) \log p\) for all primes \(p\). This paper extends results of the second author [ibid. 8, 40--42 (1976; Zbl 0321.10002)] where differences, rather than linear combinations, were considered.