On the exponential sum-product problem
Let \(p\) be a large prime and let \(g\) be an element of \(\mathbb F_p\) of a (large) multiplicative order \(T\). Given sets \(\mathcal A, \mathcal B \subset \mathbb Z_T^*\), consider the sets \[ \mathcal U = \{ g^{ab} : a \in \mathcal A, b \in \mathcal B \}, \quad \mathcal V = \{ g^{a} + g^{b} : a \in \mathcal A, b \in \mathcal B \}. \] In this paper, the author obtains non-trivial lower bounds for \(\mu = \max( \#\mathcal U, \#\mathcal V)\). Suppose that \(\#\mathcal B \leq \#\mathcal A \leq T^{1-\delta}\) for some \(\delta > 0\). When \(T \geq p^{1/2 + \varepsilon}\) for some fixed \(\varepsilon > 0\), the author uses an exponential sum estimate due to \textit{M. Z. Garaev} and \textit{A. A. Karatsuba} [Arch. Math. 87, No. 1, 33--40 (2006; Zbl 1175.11045)] to give an explicit lower bound for \(\mu\) in terms of \(p, T, \#\mathcal A\) and \(\#\mathcal B\). In particular, when \(\#\mathcal A\) and \(\#\mathcal B\) are approximately equal and \(T\) is close to \(p\), that bound is non-trivial provided that \(\#\mathcal A \geq T^{3/4 - \delta}\). For smaller values of \(T\), the author obtains another, more qualitative result. Drawing on work by \textit{J. Bourgain} [Geom. Funct. Anal. 15, No. 1, 1--34 (2005; Zbl 1102.11041)], he shows that if \(T \geq p^{\epsilon}\) for some fixed \(\varepsilon > 0\), one has \(\mu \geq \#\mathcal Ap^{\kappa}\) for some \(\kappa > 0\), provided that \(\#\mathcal A \geq T^{\eta + \delta}\) for some \(\eta \in (0, 1)\).
scientific article; zbMATH DE number 5526746
scientific article