It was shown by Carbery, Christ, and Wright that any measurable set E in the unit square in R² not containing the corners of a rectangle with area greater than λ has measure bounded by O(\sqrt{λlog(1/λ)}). We remove the log under the additional assumption that the set does not contain the corners of any axis-parallel, possibly self-crossing hexagon with unsigned area bigger than λ. Our proof may be viewed as a bilinearization of Carbery, Christ, and Wright's argument.

The author was supported by EPSRC GR/l10024