Prove the triple-sum dimension conjecture for arbitrary Borel subsets of the parabola

Prove that \(\dim_{\mathrm H}(A+A+A)=3\dim_{\mathrm H}(A)\) for arbitrary Borel subsets \(A\) of the parabola whenever the expression is expected to be valid, extending the result currently established only for Ahlfors-regular subsets.

Background

The paper proves the triple-sum dimension identity for Ahlfors-regular subsets of the parabola with dimension at most $1/2$. It explicitly states that the same equality is conjectured for arbitrary Borel subsets of the parabola and identifies the Ahlfors-regular theorem as the first non-trivial class for which the conjecture has been verified.

References

This is conjectured to hold for arbitrary Borel subsets of the parabola, and this is the first non-trivial class of sets for which it has been verified.

— Spread-out Furstenberg sets I: the sticky case  (2610.13677 - Orponen et al., 9 Oct 2026) in Section 1, subsection “Applications”