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I have seen a few papers using the delta symbol method (after Heath-Brown's A New Form of the Circle Method and Duke-Friedlander-Iwaniec's paper), where one creatively expands the delta symbol. Are there some examples, where this has been used successfully in the context of number fields?

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Yes. See the paper:

  • Browning, T. D.; Vishe, P. Cubic hypersurfaces and a version of the circle method for number fields, Duke Math. J. 163 (2014), no. 10, 1825–1883, doi:10.1215/00127094-2738530, arXiv:1207.2385.
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