Grokability in five inequalities

Recent work shows that LLM agents can improve sharp-constant inequalities by searching for extremal constructions, which yield upper bounds. We address the complementary side: a lower bound holds for every admissible function and follows from a convex relaxation of the nonconvex problem, with tighter relaxations giving stronger bounds. We instantiate the autoresearch paradigm to discover such relaxations: a coding agent proposes valid tightening constraints, a theory agent verifies each one and searches for counterexamples, and every reported bound is certified by an explicit dual-feasible point checked in rigorous interval arithmetic. On two optimization constants studied by \citet{tao2025alphaevolve} - the first autocorrelation inequality ($C_{6.2}$) and the Erdős minimum-overlap constant ($C_{6.5}$) - we improve the certified lower bounds from $1.28$ to $1.2937$ and from $0.379005$ to $0.37912$, respectively.

Let $t(N)$ be the largest $t$ for which there exist distinct sets $A_1,\dots,A_t \subseteq \{1,\dots,N\}$ such that $A_i \cap A_j$ is a nonempty arithmetic progression for all $i \neq j$ (Erdos Problem #272). Simonovits and Sos proved $t(N)=O(N^2)$ and conjectured $\binom{N}{2}+1$ is best possible; Szabo disproved this by a construction giving $t(N) \geq \binom{N}{2}+1+\lfloor(N-1)/4\rfloor$, proved the asymptotics $t(N)=N^2/2+O(N^{5/3}(\log N)^3)$, and asked whether $t(N)=\binom{N}{2}+O(N)$ and whether some element lies in all sets of any extremal family (the kernel question). We determine $t(N)$ exactly for all $3 \leq N \leq 12$ by exhaustive computation: in this entire range Szabo's lower bound is exact, and we conjecture that $t(N)=\binom{N}{2}+1+\lfloor(N-1)/4\rfloor$ for every $N$. Towards the matching upper bound we prove, for every $N$, that Szabo's bound is the exact maximum over all families with a common element (starred families). The proof combines a self-contained ``defect-one'' counting inequality for staircase regions with a new structural theorem: every non-progression member of such a family contains a bad pair that no other member can share. Consequently the sharpened conjecture reduces to a single remaining statement, namely Szabo's kernel conjecture that some element lies in all sets of an extremal family, and we prove first structural constraints on putative non-starred extremal families.

Carbery proposed the following sharpened form of triangle inequality for many functions: for any $p\ge 2$ and any finite sequence $(f_j)_j\subset L^p$ we have $$\Big\|\sum_j f_j\Big\|_p \ \le\ \left(\sup_{j} \sum_{k} α_{jk}^{\,c}\right)^{1/p'} \Big(\sum_j \|f_j\|_p^p\Big)^{1/p},$$ where $c=2$, $1/p+1/p'=1$, and $α_{jk}=\sqrt{\frac{\|f_{j}f_{k}\|_{p/2}}{\|f_{j}\|_{p}\|f_{k}\|_{p}}}$. 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 $c\le p'$. Finally, at the critical exponent $c=p'$, we establish the inequality for all integer values $p\ge 2$. In the second part of the paper we obtain a sharp three-function bound $$\Big\|\sum_{j=1}^{3} f_j\Big\|_p \ \le\ \left(1+2Γ^{c(p)}\right)^{1/p'} \Big(\sum_{j=1}^{3} \|f_j\|_p^p\Big)^{1/p},$$ where $p \geq 3$, $c(p) = \frac{2\ln(2)}{(p-2)\ln(3)+2\ln(2)}$ and $Γ=Γ(f_1,f_2,f_3)\in[0,1]$ quantifies the degree of orthogonality among $f_1,f_2,f_3$. The exponent $c(p)$ is optimal, and improves upon the power $r(p) = \frac{6}{5p-4}$ 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.