• Updates on my research and expository papers, discussion of open problems, and other maths-related topics. • By Terence Tao Home About Career advice On writing Books Mastodon+ Applets Subscribe to feed Polynomial towers and inverse Gowers theory for bounded-exponent groups 6 January, 2026 inmath.CO,math.DS,paper| Tags:Asgar Jamneshan,Gowers-Host-Kra seminorms,inverse conjecture for the Gowers norm,Or Shalom| byTerence Tao Asgar Jamneshan,Or Shalomand I have uploaded to the arXiv our paper “Polynomial towers and inverse Gowers theory for bounded-exponent groups”. • This continues our investigation into the ergodic-theory approach to the inverse theory of Gowers norms over finite abelian groups. • In this regard, our main result establishes a satisfactory (qualitative) inverse theorem for groupsof bounded exponent: Theorem 1Letbe a finite abelian group of some exponent, and letbe-bounded with. • Then there exists a polynomialof degree at mostsuch that This type of result was previously known in the case of vector spaces over finite fields (by workof myselfand Ziegler), groups of squarefree order (by workof Candela, González-Sánchez, and Szegedy), and in thecase (by work ofJamneshan and myself). • The case, for instance, is treated by this theorem but not covered by previous results.
Article Summaries:
- A new arXiv paper by Jamneshan, Shalom, and the author presents a qualitative inverse theorem for Gowers norms on finite abelian groups of bounded exponent. The result shows that any bounded‑valued function with large Uⁿ‑norm correlates with a phase polynomial of degree ≤ n−1, extending earlier work that handled only vector spaces over finite fields, square‑free groups, or groups of exponent p. Unlike previous proofs, the authors avoid separating high‑ and low‑characteristic cases, using an ergodic‑theory approach based on a Host-Kra structure theorem. They prove a weaker Abramov‑type extension, then use an algebraic splitting argument to recover the full correlation on the original group.
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