Almost-Orthogonality in Lp Spaces: A Case Study with Grok
Authors: Ziang Chen, Jaume de Dios Pont, Paata Ivanisvili, Jose Madrid, Haozhu Wang
Abstract
Carbery proposed the following sharpened form of triangle inequality for many functions: for any p≥2 and any finite sequence (fj)j⊂Lp we have
j∑fjp≤(jsupk∑αjkc)1/p′(j∑∥fj∥pp)1/p,
where c=2, 1/p+1/p′=1, and αjk=∥fj∥p∥fk∥p∥fjfk∥p/2. In the first part of this paper we construct a counterexample showing that this inequality fails for every p>2. We then prove that if an estimate of the above form holds, the exponent must satisfy c≤p′. Finally, at the critical exponent c=p′, we establish the inequality for all integer values p≥2. In the second part of the paper we obtain a sharp three-function bound
j=1∑3fjp≤(1+2Γc(p))1/p′(j=1∑3∥fj∥pp)1/p,
where p≥3, c(p)=(p−2)ln(3)+2ln(2)2ln(2) and Γ=Γ(f1,f2,f3)∈[0,1] quantifies the degree of orthogonality among f1,f2,f3. The exponent c(p) is optimal, and improves upon the power r(p)=5p−46 obtained previously by Carlen, Frank, and Lieb. Some intermediate lemmas and inequalities appearing in this work were explored with the assistance of the large language model Grok.
In this note, we report five mathematical discoveries made in collaboration with Grok, all of which have been subsequently verified by the authors. These include an improved lower bound on the maximal Gaussian perimeter of convex sets in Rn, sharper L2-L1 moment comparison inequalities on the Hamming cube {−1,1}n, a strengthened autoconvolution inequality, improved asymptotic bounds on the size of the largest g-Sidon sets in {1,…,n}, and an optimal balanced Szarek's inequality.
We prove function-theoretic analogues of a quantitative result of Hodges on extracting the order property from a sufficiently large 2-tree coded in a binary relation. Similar analogues for functions were previously obtained by Daskalakis and Golowich and by Anderson and Benedikt. These results are from statistical learning theory, where 2-trees are captured by sequential fat-shattering dimension, and the order property is controlled by various notions of "thresholds". Our first main result (Theorem 1.11) focuses on extracting a less restrictive kind of threshold from a tree, and yields significantly better bounds compared to what can be obtained from earlier results focusing on more restrictive versions. Part of the motivation for Theorem 1.11 lies in a companion paper, where this theorem is used to obtain efficient bounds in quantitative regularity lemmas for "stable functions". Here will use Theorem 1.11 to reprove a result of Anderson and Benedikt in a stronger form and with improved bounds. We also use Theorem 1.11 to prove an at most double-exponential bound on dual sequential fat-shattering, which resolves an open problem. In our second main result (Theorem 1.14), we give a new proof of a result of Daskalakis and Golowich on extracting "tight thresholds" from large sequential fat-shattering dimension, with improved bounds. This resolves another open problem related to correcting the proof of a result claimed by Jung, Kim, and Tewari.
In this paper, we establish a novel connection between the metric entropy growth and the embeddability of function spaces into reproducing kernel Hilbert/Banach spaces. Metric entropy characterizes the information complexity of function spaces and has implications for their approximability and learnability. Classical results show that embedding a function space into a reproducing kernel Hilbert space (RKHS) implies a bound on its metric entropy growth. Surprisingly, we prove a \textbf{converse}: a bound on the metric entropy growth of a function space allows its embedding to a Lp−type Reproducing Kernel Banach Space (RKBS). This shows that the Lp−type RKBS provides a broad modeling framework for learnable function classes with controlled metric entropies. Our results shed new light on the power and limitations of kernel methods for learning complex function spaces.