Equations in finite fields with restricted solution sets. I: Character sums
Let \(p\) be a prime number, \(r\) a positive integer, and \(\mathbb F_q\) a finite field with \(q=p^r\) elements. \textit{K.~Gyarmati} [Acta Arith. 97, 53--65 (2001; Zbl 0986.11016)] showed that if \(h(x)\in \mathbb F_{p}[x]\) and \(A,B\) are subsets of \(\mathbb F_p\) such that \(| A| | B| >cp\), where \(c\) is a positive constant depending on the degree of \(h(x)\), then there exist elements \(a\in A, b\in B\) and \(x\in \mathbb F_p\) with \(a+b=h(x)\) (resp. \(ab=h(x)\)). Later, \textit{A.~Sárközy} [Acta Arith. 118, 403--409 (2005; Zbl 1078.11011)] proved that if \(A,B,C,D\in \mathbb F_p\) are subsets of \(\mathbb F_p\) such that \(| A| | B| | C| | D| >p^3\) (resp. \(| A| | B| | C| | D| >100p^3\)) then there exist \(a\in A\), \(b\in B\), \(c\in C\), \(d\in D\) with \(a+b=cd\) (resp. \(ab+1=cd\)). In a series of papers the authors intend to study equations of the above type over an arbitrary finite field \(\mathbb F_q\). In this paper (which presents Part I of the series) the authors give the necessary character sum estimates which generalize several estimates obtained earlier in the case of prime finite field \(\mathbb F_p\) by \textit{I.~M. Vinogradov} [Tr. Mat. Inst. Steklova 23, 110 pp. (1947; Zbl 0041.37002)], \textit{P.~Erdős} and \textit{H.~N. Shapiro} [Pac. J. Math. 7, 861--865 (1957; Zbl 0079.06304)], \textit{J.~Friedlander} and \textit{H.~Iwaniec} [Proc. Am. Math. Soc. 119, 365--372 (1993; Zbl 0782.11022)]. For Part II, see Acta Math. Hung. 119, No. 3, 259--280 (2008; Zbl 1199.11141).
scientific article; zbMATH DE number 5355076