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A bilinear version of Bogolyubov's theorem | | W. T. Gowers
; L. Milićević
; | | Date: |
1 Dec 2017 | | Abstract: | A theorem of Bogolyubov states that for every dense set $A$ in $mathbb{Z}_N$
we may find a large Bohr set inside $A+A-A-A$. In this note, motivated by the
work on a quantitative inverse theorem for the Gowers $U^4$ norm, we prove a
bilinear variant of this result in vector spaces over finite fields. Namely, if
we start with a dense set $A subset mathbb{F}^n_p imes mathbb{F}^n_p$ and
then take rows (respectively columns) of $A$ and change each row (respectively
column) to the set difference of it with itself, repeating this procedure
several times, we obtain a bilinear analogue of a Bohr set inside the resulting
set, namely the zero set of a biaffine map from $mathbb{F}^n_p imes
mathbb{F}^n_p$ to a $mathbb{F}_p$-vector space of bounded dimension. An
almost identical result was proved independently by Bienvenu and L^e. | | Source: | arXiv, 1712.0248 | | Services: | Forum | Review | PDF | Favorites | |
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