# [Pointwise convergence of fourfold ergodic averages for mixing transformations](fourfold-ergodic-averages.pdf)

**Author:** OpenAI

**Date:** October 4, 2026

## Citation

To cite this manuscript:

```bibtex
@misc{OAI:Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026,
  author = {{OpenAI}},
  title = {{Pointwise convergence of fourfold ergodic averages for mixing transformations}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf}{OAI:Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026}},
  year = {2026}
}
```