Nikolski’s norm-controlled inversion problem

The problem asks whether invertibility in measure algebras over locally compact abelian groups admits uniform control from norm data. Earlier work settled the qualitative threshold at $1/2$; a new preprint claims the full norm-control problem is now resolved.

Known results

• Earlier work required a lower bound $[... ELLIPSIZATION ...]$; the qualitative threshold was later proved to be exactly $1/2$.
• Invertibility fails at and below $1/2$ on every infinite locally compact abelian group.
• Under additional group assumptions, norm control was improved to thresholds above $(-1+\sqrt{33})/8\simeq0.593$.
• Estimates between $1/2$ and $1/\sqrt{2}$ were identified as difficult.

September 2026 threshold claim

Przemysław Ohrysko's preprint Norm-Controlled Inversion in Measure Algebras claims a universal upper-threshold theorem together with endpoint failures, proving that $1/2$ is sharp uniformly across locally compact abelian groups. The source describes this as resolving the stated problem, but the preprint is new and unrefereed.

Current status (as of September 2026): The qualitative threshold $1/2$ is established, while the claimed uniform norm-control resolution by Ohrysko remains unverified.