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On sums and products of residues modulo \(p\). (English) Zbl 1078.11011

Let \(p\) be a prime number and let \(A,~B,~C,~D\) be subsets of \(\mathbb Z_p\). Write \(N\) for the number of solutions of the equation \[ a+b=cd\quad a\in A,~b\in B,~c\in C,~d\in D.\tag{1} \] Using exponential sums, the author shows that \[ \Big| N-{{| A| | B| | C| | D| }\over {p}}\Big| \leq (| A| | B| | C| | D| )^{1/2}p^{1/2}. \] In particular, equation (1) admits a solution if \(| A| | B| | C| | D| >p^3\). The author deduces several nice consequences of his result, an example of which is that if \(p\) is an odd prime and \(A\) and \(B\) are subsets of \({\mathbb Z}_p\) with \(| A| | B| >4(1-1/p)^{-2}p,\) then there are \(a,~a'\) in \(A\) and \(b,~b'\) in \(B\) such that \(a+b\) is a quadratic residue modulo \(p\) and \(a'+b'\) is not.

MSC:

11B30 Arithmetic combinatorics; higher degree uniformity
11L07 Estimates on exponential sums
11B50 Sequences (mod \(m\))
11T23 Exponential sums
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