A counterexample to the Gowers U^k uncertainty conjecture for k≥ 6

@misc{vranesko2026counterexample,
  title = {{A counterexample to the Gowers $U^k$ uncertainty conjecture for $k\ge 6$}},
  author = {Iana Vranesko},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06531},
  eprint = {2609.06531},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06531},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - A counterexample to the Gowers U^k uncertainty conjecture for k≥ 6
AU  - Iana Vranesko
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06531
UR  - https://arxiv.org/abs/2609.06531
AB  - The Donoho--Stark uncertainty principle asserts N^d≤|E| |Σ|, and the additive energy uncertainty principle of Aldahleh--Iosevich--Iosevich--Jaimangal--Mayeli--Pack refines it to 1≤|E|·|1_Σ|_U^2^4/3. It has been conjectured that the natural higher-order analogue 1≤ |E|·|1_Σ|_U^k^2^k/(k+1) holds for every k≥ 2, the coset case giving equality. We show that this fails for every k≥ 6. The counterexample is the two-point signal f=δ_0−δ_1 on ℤ_3, for which the left-hand quantity equals 2/3(2(k+1))^1/(k+1); this is <1 precisely when k≥ 6, and tends to 2/3.
ER  -