The equation \(x_1x_2=x_3x_4+\lambda \) in fields of prime order and applications
The congruence \(x_1 x_2 \equiv x_3 x_4 \pmod p\), where \(p\) is a large prime, arises in many problems of number theory studied, e.g., by \textit{J. Bourgain, N. Katz} and \textit{T. Tao} [Geom. Funct. Anal. 14, No. 1, 27--57 (2004; Zbl 1145.11306)], by \textit{J. B. Friedlander} and \textit{H. Iwaniec} [Acta Arith. 45, 273--277 (1985; Zbl 0572.10033)], by \textit{G. Harman} [Math. Proc. Camb. Philos. Soc. 95, 381--388 (1984; Zbl 0539.10025)], by \textit{H. L. Montgomery} and \textit{R. C. Vaughan} [Can. J. Math. 31, 476--487 (1979; Zbl 0416.10030)] and by \textit{R. C. Vaughan} [Proc. Lond. Math. Soc., III. Ser. 33, 177--192 (1976; Zbl 0327.10031)]. In this paper new results related to the congruence \(x_1 x_2 \equiv x_3 x_4 +\lambda \pmod p\), where \(\lambda \) is an integer, are obtained as a combination of methods developed by \textit{A. Ayyad, T. Cochrane} and \textit{Z. Zheng} [J. Number Theory 59, No. 2, 398--413 (1996; Zbl 0869.11003)], by \textit{J. Bourgain, A. A. Glibichuk} and \textit{S. V. Konyagin} [J. Lond. Math. Soc., II. Ser. 73, No. 2, 380--398 (2006; Zbl 1093.11057)], by \textit{D. A. Burgess} [Mathematika 33, 1--5 (1986; Zbl 0595.10028)], by \textit{T. Cochrane} and \textit{Z. Zheng} [Acta Math. Sin., New Ser. 14, No.2, 175--182 (1998; Zbl 0912.11015)], by \textit{J. Friedlander} and \textit{H. Iwaniec} [Proc. Am. Math. Soc. 119, No.2, 365--372 (1993; Zbl 0782.11022)] and by the first author \textit{M. Z. Garaev} [Acta Arith. 124, No. 1, 27--39 (2006; Zbl 1158.11002)]. In their theorem-proving the authors employ trigonometric sums, combinatorial arguments, the \textit{Cauchy-Schwarz inequality}, the \textit{pigeon-hole principle} and they recall previous works by \textit{M. Z. Garaev} and \textit{A. A. Karatsuba} [Proc. Edinb. Math. Soc., II. Ser. 50, No. 2, 363--375 (2007; Zbl 1197.11003)], by \textit{A. A. Glibichuk} [Math. Notes 79, No. 3, 356--365 (2006); translation from Mat. Zametki 79, No. 3, 384--395 (2006; Zbl 1129.11004)], by \textit{I. E. Shparlinski} [Number theory. Sailing on the sea of number theory. Proceedings of the 4th China-Japan seminar on number theory, Weihai, China, August 30--September 3, 2006. Hackensack, NJ: World Scientific. Series on Number Theory and its Applications 2, 155--189 (2007; Zbl 1175.11044)], by \textit{I. E. Shparlinski} [Mich. Math. J. 56, No. 1, 99--111 (2008; Zbl 1225.11101)] and especially by \textit{G. Tenenbaum} [Compos. Math. 51, 243--263 (1984; Zbl 0541.10038)].
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