A Problem on Linear Functions and Subsets of a Finite Field

Let \(g: \mathbb{F}^m \to \mathbb{F}\) be a linear function on the vector space \(\mathbb{F}^m\) over a finite field \(\mathbb{F}\). A subset \(S \subsetneqq \mathbb{F}\) is called \(g\)-thin iff \(g(S^m) \subsetneqq \mathbb{F}\). In case \(\mathbb{F}\) is the field \(\mathbb{Z}_p\) of odd prime order, if \(S\) is \(g\)-thin and if \(m\) divides \(p-1\), then it is shown that \(|S| \leq \frac{p-1}{m}\). We also show that in certain cases \(S\) must be an arithmetic progression, and the form of the linear function \(g\) can be characterized.