Talagrand’s convolution tail bound on the Boolean cube
Follow the latest progress by email.
On the uniform Boolean cube, does convolution by a fixed biased-coin product measure improve Markov’s tail bound by order 1/√log u for every nonnegative L¹-normalized function, with a constant independent of dimension?
Why it matters and what counts as progress
Why it matters: The target is sharp dimension-free tail regularization by a simple noise operator.
What counts as progress: A bound sup_f u·λ({T_μₐf≥u})≤Cₐ/√log u for every n≥1, u>1 and fixed 0<a<1, without an additional log-log loss.
How the problem developed
- 1989 · Problem [1]
Michel Talagrand Asked whether convolution regularization of L¹ functions yields tails uniformly better than Markov’s inequality.
- 13 Aug 2014 · Gaussian case [2]
Ronen Eldan and James Lee Reported a positive answer to the Gaussian-space form of Talagrand’s convolution question.
- 24 Nov 2025 · Partial bound [3]
Yuansi Chen Proved the Boolean-cube tail bound up to a dimension-free log log η factor using a perturbed reverse-heat coupling.
- 16 Aug 2026 · Sharp bound [4]
Odin Automatic AI Research Agent Proved the exact Boolean-cube weak-type bound ψμₐ(u)≤Cₐ/√log u for every u>1 and dimension n, removing the iterated-log factor.
AI contribution
Weak-Type Bounds for Convolution on the Boolean Hypercube
On the Boolean cube {−1,1}^n with uniform measure, for every fixed 0<a<1, n≥1 and u>1, convolution by the biased-coin product measure μ_a satisfies sup_{f≥0, ||f||₁=1} u·λ({T_{μ_a}f≥u})≤C_a/√(log u), with an a-dependent constant. The bound is dimension free.
Sources
- Michel Talagrand, Regularization of L¹ functions by convolution
- Microsoft Research, Talagrand’s Convolution Conjecture
- Yuansi Chen, Talagrand’s convolution conjecture up to loglog via perturbed reverse heat
- Junwei Lu, Shengtao Guo, and Ethan X. Fang, Weak-Type Bounds for Convolution on the Boolean Hypercube
- Original convolution theorem and proof
- Talagrand review
- Later independent martingale proof
Website timeline
Publication dates and editorial changes.