Talagrands Convolution Conjecture

### What is the conjecture

**Conjecture:** For every $a > 0$, there exists a constant $C(a) > 0$ such that for all $u \geq 2$,
$$\psi_{\mu_a}(u) \leq \frac{C(a)}{\sqrt{\log u}}$$

This is complemented by Proposition 5, which establishes a matching lower bound $\psi_{\mu_a}(u) \geq C'(a)/\sqrt{\log u}$ for $u \geq 2$. Together, these would establish that the optimal decay rate of $\psi_{\mu_a}(u)$ is precisely $1/\sqrt{\log u}$.

(This description may contain subtle errors especially on more complex problems; for exact details, refer to the sources.)

A proof is worth 1000$.

**Sources:**
- https://michel.talagrand.net/prizes/convolution/convolution.tex

### Prerequisites needed

**Formalizability Rating:** 4/5 (0 is best) (as of 2026-01-22)

Building blocks (1-3; from search results):
- Product measures on discrete spaces ($\{-1,1\}^{\mathbb{N}}$) and probability measure basics
- Convolution operators and $L^1$ norm theory (partially available in Mathlib)
- Supremum over function spaces and measure-theoretic tail bounds

Missing pieces (exactly 2; unclear/absent from search results):
- Formal definition and theory of convolution operators on infinite product spaces and their tail behavior under specific probability distributions
- Complete framework for handling supremum-based functionals over sets of measures with specified properties, particularly for non-standard convolution contexts

Rating justification: The statement requires measure theory and functional analysis, both of which have substantial Mathlib support. However, the specific functional $\psi_\mu$ and the convolution operator $T_\mu$ in this particular measure-theoretic setting are not standard Mathlib definitions, requiring moderate additional infrastructure to state formally.

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

* ams-60
* ams-28
* ams-42

### Choose either option

- [ ] I plan on adding this conjecture to the repository
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

---
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Read the linked convolution.tex source first, then inspect the existing Mathlib support for product measures, convolution operators, L1 norms, and measure-theoretic bounds. The issue identifies missing infrastructure for the functionals ψμ and Tμ on infinite product spaces, so completion would require a formal statement of the conjecture together with the needed definitions and supporting theory.

Written by the indexing model from the issue text.