Convergence of the limited augmented Zarankiewicz density below quadratic growth

Determine whether the normalized limited augmented Zarankiewicz parameter zL(m,n)/(mn) converges when n≤m=o(n²), including the square case m=n, and establish whether its limiting value is common throughout this regime or depends on the growth rate of m relative to n.

Background

Theorem 1 bounds the normalized limited parameter in the regime n≤m=o(n²) between asymptotic coefficients 1/4 and 3/10, but does not identify a limiting density. The paper specifically leaves unresolved whether convergence occurs even for square matrices and whether the limit, if it exists, is universal across the entire subquadratic-growth regime or varies with the aspect-ratio growth.

References

Does the normalized limited parameter converge even in the square case m = n? More generally, is there a common limiting constant throughout this regime, or does the answer depend on the growth of m relative to n?

— Density and separation for augmented Zarankiewicz numbers  (2609.16555 - Lebedev, 15 Sep 2026) in Section 10, “The limited density below quadratic growth,” p. 23

For fixed 0 < c < 1/12, does zL(m, n)/(mn) have a limit depending only on c whenever m/n2 → c? Theorem 15 separates it from 1/3, while Theorem bounds the loss on both sides near the threshold. An exact profile would also determine the leading coefficient of that loss as c ↑ 1/12.

— Density and separation for augmented Zarankiewicz numbers  (2609.16555 - Lebedev, 15 Sep 2026) in Section 10, “The density profile below the threshold,” p. 23