Abstract
We obtain smoothing estimates for certain nonlinear convolution operators on prime fields, leading to quantitative nonlinear Roth type theorems. For instance, we produce triplets x, x + y, x + y 2 and x, x + y, x + \(\bar y\) in proportional subsets of F p .
Compared with the usual linear setting (i.e. arithmetic progressions), the nonlinear nature of the operators leads to different phenomena, both qualitatively and quantitatively. The methods used are purely analytical.
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The authors were partially supported by NSF grants DMS-1301619 and DMS-1600154.
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Bourgain, J., Chang, MC. Nonlinear Roth type theorems in finite fields. Isr. J. Math. 221, 853–867 (2017). https://doi.org/10.1007/s11856-017-1577-9
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DOI: https://doi.org/10.1007/s11856-017-1577-9