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Abstract  

Generalizing earlier results, it is shown that if
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}$$ \end{document}
are “large” subsets of a finite field F q, then the equations a + b = cd, resp. ab + 1 = cd can be solved with
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$a \in \mathcal{A}, b \in \mathcal{B}, c \in \mathcal{C}, d \in \mathcal{D}$$ \end{document}
. Other algebraic equations with solutions restricted to “large” subsets of F q are also studied. The proofs are based on character sum estimates proved in Part I of the paper.
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Abstract  

In earlier papers, for “large” (but otherwise unspecified) subsets A, B of Z p and for h(x) ∈ Z p[x], Gyarmati studied the solvability of the equations a + b = h(x), resp. ab = h(x) with aA, bB, xZ p, and for large subsets A, B, C, D of Z p Sárközy showed the solvability of the equations a + b = cd, resp. ab + 1 = cd with aA, bB, cC, dD. In this series of papers equations of this type will be studied in finite fields. In particular, in Part I of the series we will prove the necessary character sum estimates of independent interest some of which generalize earlier results.

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’s functional equations and approximate algebra homomorphisms, Bull. Korean Math. Soc. , 39 (2002), no. 3, 401–410. MR 2004g :39047 Park W.-G. Generalized Jensen’s functional

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