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On the Fourier Entropy-Influence Conjecture for Boolean Plateaued Functions

  • Vladimir N. Potapov
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Abstract

We prove the following inequality for Boolean functions: 2∑x∈ F2nf(x)wt(x)≥ wt(f)(n−deg(f)). Using this inequality, we establish the Fourier Entropy-Influence (FEI) conjecture for Boolean plateaued functions. In particular, we show that the sharp FEI constant for the class of plateaued functions is 4. We also prove the FEI conjecture for partially bent functions and show that the corresponding sharp constant is 2. Finally, we derive several estimates for the p-biased distribution on the Boolean hypercube. Keywords: Fourier entropy, total influence, average sensitivity, plateaued function, algebraic degree, Reed-Muller code, p-biased distribution.

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Cite

BibTeXRIS
BibTeX
@misc{potapov2026fourier,
  title = {{On the Fourier Entropy-Influence Conjecture for Boolean Plateaued Functions}},
  author = {Vladimir N. Potapov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21563},
  eprint = {2609.21563},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21563},
}
RIS
TY  - UNPB
TI  - On the Fourier Entropy-Influence Conjecture for Boolean Plateaued Functions
AU  - Vladimir N. Potapov
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21563
UR  - https://arxiv.org/abs/2609.21563
AB  - We prove the following inequality for Boolean functions: 2∑_x∈ F_2^nf(x)wt(x)≥ wt(f)(n−deg(f)). Using this inequality, we establish the Fourier Entropy-Influence (FEI) conjecture for Boolean plateaued functions. In particular, we show that the sharp FEI constant for the class of plateaued functions is 4. We also prove the FEI conjecture for partially bent functions and show that the corresponding sharp constant is 2. Finally, we derive several estimates for the p-biased distribution on the Boolean hypercube. Keywords: Fourier entropy, total influence, average sensitivity, plateaued function, algebraic degree, Reed-Muller code, p-biased distribution.
ER  -