Geodesic
On congruences with products of variables from short intervals and applications
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Orthogonal series, approximation theory, and related problems, Tome 280 (2013), pp. 67-96

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We obtain upper bounds on the number of solutions to congruences of the type $(x_1+s)\dots(x_\nu+s)\equiv(y_1+s)\dots(y_\nu +s)\not\equiv0\pmod p$ modulo a prime $p$ with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M. Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. Friedlander and H. Iwaniec and some results of M.-C. Chang and A. A. Karatsuba on character sums twisted with the divisor function. 
@article{TM_2013_280_a4,
     author = {Jean Bourgain and Moubariz Z. Garaev and Sergei V. Konyagin and Igor E. Shparlinski},
     title = {On congruences with products of variables from short intervals and applications},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {67--96},
     publisher = {mathdoc},
     volume = {280},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_280_a4/}
}
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Jean Bourgain; Moubariz Z. Garaev; Sergei V. Konyagin; Igor E. Shparlinski. On congruences with products of variables from short intervals and applications. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Orthogonal series, approximation theory, and related problems, Tome 280 (2013), pp. 67-96. http://geodesic.mathdoc.fr/item/TM_2013_280_a4/