Kakeya implications for genuinely irrational mixed-norm projections

Determine whether the mixed-norm BrascampLieb inequality with q<2 and 1/q+1/r=1 has implications for Kakeya sets when the projections B_j cannot be simultaneously made rational under affine transformations.

Background

The paper establishes a connection between mixed-norm BrascampLieb inequalities for rational projections and Hausdorff-dimensional Kakeya estimates. It notes that the behavior of the inequality is less intuitive for genuinely irrational projections, defined as projections that cannot be simultaneously well approximated by rationals in an appropriate sense, and explicitly leaves open whether the Kakeya connection persists in that setting.

References

Assume assumptionofqandr. Does BLmix for $q<2$ have implications on Kakeya if $B_j$'s cannot be simultaneously made rational under affine transformations?

assumptionofqandr:

1q+1r=1.\frac{1}{q}+ \frac{1}{r} = 1.

BLmix:

∥∏j=1mfj(Bj(x,y))pj∥LxqLyr≤C∏j=1m∥fj∥L1pj\|\prod_{j=1}^m f_j (B_j (x, y))^{p_j}\|_{L_x^q L_y^r} \leq C \prod_{j=1}^m \|f_j\|_{L^1}^{p_j}

— Mixed-norm Brasamp-Lieb inequalities  (2608.17952 - Zhang, 18 Aug 2026) in Section 5.2, subsection “The connection to (BLmix),” Question 5.5