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An Improved Exponent for Products of Differences over Arbitrary Finite Fields

  • Yao Zhi
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Abstract

Murphy and Petridis proved that, for subsets A of an arbitrary finite field Fq, the condition abs A>q2/3−δ implies abs(A−A)(A−A)>q/2 whenever δ<1/13542 and q is sufficiently large. We improve the explicit saving to 1/1689. The proof replaces the main physical-space extraction in their large-energy branch by a common Fourier spectrum, followed by a popular-ratio Balog--Szemerédi--Gowers step. The resulting additive energy is treated level by level, retaining difference multiplicities and combining the Reiher--Schoen relation lemma with Mohammadi's difference--ratio estimate. This gives the structural exponent $188$ and hence abs A≥ Cq375/563=Cq2/3−1/1689 Longrightarrow abs(A−A)(A−A)>q/2 for an absolute constant C>0.

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Cite

BibTeXRIS
BibTeX
@misc{zhi2026improved,
  title = {{An Improved Exponent for Products of Differences over Arbitrary Finite Fields}},
  author = {Yao Zhi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.38244},
  eprint = {2609.38244},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38244},
}
RIS
TY  - UNPB
TI  - An Improved Exponent for Products of Differences over Arbitrary Finite Fields
AU  - Yao Zhi
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.38244
UR  - https://arxiv.org/abs/2609.38244
AB  - Murphy and Petridis proved that, for subsets A of an arbitrary finite field F_q, the condition abs A>q^2/3−δ implies abs(A−A)(A−A)>q/2 whenever δ<1/13542 and q is sufficiently large. We improve the explicit saving to 1/1689. The proof replaces the main physical-space extraction in their large-energy branch by a common Fourier spectrum, followed by a popular-ratio Balog--Szemerédi--Gowers step. The resulting additive energy is treated level by level, retaining difference multiplicities and combining the Reiher--Schoen relation lemma with Mohammadi's difference--ratio estimate. This gives the structural exponent $188$ and hence abs A≥ Cq^375/563=Cq^2/3−1/1689 Longrightarrow abs(A−A)(A−A)>q/2 for an absolute constant C>0.
ER  -