On a variant of sum-product estimates and explicit exponential sum bounds in prime fields

By Jean Bourgain, Moubariz Z. Garaev. Mathematical Proceedings of the Cambridge Philosophical Society, 2009

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An explicit incidence theorem in p (2010, background): “Garaev [6] succeeded in obtaining a quantitative sumproduct estimate in Fp: for a small enough (say |A| √ p) subset A of Fp, either A+A or A·A has cardinality at least |A|1+ 114 , up to a multiple of a power of c log |A|.”
On asymptotic formulae in some sum–product questions (2018, background): “It follows that for any eigenfunction ~v, ~v 6= ~u the multiplicity of the correspondent eigenvalue is at least (p− 1)/2 (see details in [44], [10], [21]).”
New sum-product type estimates over finite fields (2014, background): “Results in this direction can be found in [3], [4], [19], [20].”
On the size of the set A(A + 1) (2008, background): “This estimate and its proof consequently have been quantified and simplified in [2], [6]–[9], [11]–[15], [17].”
On sum sets of sets having small product set (2015, methodology): “By Balog–Szemerédi–Gowers Theorem [1] (see also [6]) there is a set S ′′ τ ⊆ S ′ τ , |S ′′ τ | & η|Sτ | such that |S ′′ τ /S ′′ τ | .”