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PreprintarXiv licencearXiv

Proofs of strong forms of the Bovdi--Geche universality conjecture and a complex analogue

  • Victor Bovdi
  • Ambrus Pal
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Abstract

We prove two universality theorems for neural networks consisting of a single neuron. First we prove a strong form of the Bovdi--Geche universality conjecture on neurons over finite fields, where the activation function is a certain modification of the discrete logarithm, using the Weil bound on multiplicative character sums over finite fields. Then we show a generalisation of Aizenberg's universality theorem for complex binary neurons with a weight vector independent of the target function, positively answering another question in the literature. The proof of the latter uses an equidistribution theorem for the dilates of certain maps into tori.

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Cite

BibTeXRIS
BibTeX
@misc{bovdi2026proofs,
  title = {{Proofs of strong forms of the Bovdi--Geche universality conjecture and a complex analogue}},
  author = {Victor Bovdi and Ambrus Pal},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35645},
  eprint = {2610.35645},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35645},
}
RIS
TY  - UNPB
TI  - Proofs of strong forms of the Bovdi--Geche universality conjecture and a complex analogue
AU  - Victor Bovdi
AU  - Ambrus Pal
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35645
UR  - https://arxiv.org/abs/2610.35645
AB  - We prove two universality theorems for neural networks consisting of a single neuron. First we prove a strong form of the Bovdi--Geche universality conjecture on neurons over finite fields, where the activation function is a certain modification of the discrete logarithm, using the Weil bound on multiplicative character sums over finite fields. Then we show a generalisation of Aizenberg's universality theorem for complex binary neurons with a weight vector independent of the target function, positively answering another question in the literature. The proof of the latter uses an equidistribution theorem for the dilates of certain maps into tori.
ER  -