𝒯ₙ-Configurations & Extremal Incidences

• 𝒯ₙ-configurations are combinatorial and geometric constructions that partition the plane into an N^(1/3) × N^(1/3) grid to achieve near-maximal incidences.
• The framework employs O(N^(1/3)) parameters and structuring lines to create cellular arrangements where each cell contains roughly N^(1/3) points and N^(2/3) lines.
• This universal construction method not only realizes extremal arrangements under the Szemerédi–Trotter bound but also extends to similar settings like unit circles.

$\mathcal{T}_N$-Configurations constitute a combinatorial and geometric framework for producing arrangements of $N$ points and $N$ lines in the plane with near-maximal incidences with respect to the Szemerédi–Trotter theorem. The central construction employs $O(N^{1/3})$ parameters to divide the plane into an $N^{1/3}\times N^{1/3}$ grid of "cells", where intersection properties between points determined by generically oriented "structuring lines" result in extremal incidence configurations. Any near-extremal configuration is shown to be densely related, up to an $N^{-o(1)}$ error, to a successful instance of this recipe after projective transformation (Katz et al., 2023).

1. Formal Construction and Parameterization

Fix an integer $N \gg 1$ and set $r \approx N^{1/3}$. Define sets of real "x-cuts" $A = (a_1 < \cdots < a_{r+1})$ and "y-cuts" $B = (b_1 < \cdots < b_{r+1})$, along with a generically chosen family of $r$ structuring lines $\ell_1, \ldots, \ell_r$. The intersection of these cuts yields an axis-parallel grid of cells $C_{i,j} = [a_i, a_{i+1}] \times [b_j, b_{j+1}]$ for $1 \leq i, j \leq r$. Within each vertical strip $[a_i, a_{i+1}]$, the structuring lines provide $\approx N^{2/3}$ intersection points declared as candidate points for row $i$. In each horizontal cut $B$, if every vertical strip delivers $\approx N^{1/3}$ points for at least $c \cdot r$ values of $j$, the collection is counted as "successfully built", forming the point set $P(A, B, \ell_1, \ldots, \ell_r)$. For each cell, pairwise lines between candidate points are retained in the final line set $L(A,B,\ell_1,\ldots,\ell_r)$ if they intersect $\gtrsim N^{1/3}$ structuring lines within the cell. If $c \cdot r^2$ such cells yield $\approx N^{2/3}$ lines, with each cell containing $\approx N^{1/3}$ points, the recipe is termed a success, producing a $\mathcal{T}_N$-configuration (Katz et al., 2023).

2. Extremality: Szemerédi–Trotter Incidence Bound

The Szemerédi–Trotter theorem states any configuration of $N$ points and $N$ lines achieves at most $O(N^{4/3})$ incidences, i.e., $I(N) := \#$incidences $\leq C N^{4/3} + O(N)$. Extremal configurations realize $I(N) \approx N^{4/3}$. In a successful $\mathcal{T}_N$-configuration, there are $r^2 \approx N^{2/3}$ rich cells, each containing $\approx N^{1/3}$ points and $\approx N^{2/3}$ lines. Each of these lines meets two distinct points in the same cell, resulting in a total incidence count $$r^2 \cdot (N^{1/3} \cdot 2) \approx N^{2/3} \cdot N^{1/3} \approx N^{4/3}$$ (Katz et al., 2023).

3. Structural Properties and Cell Decompositions

A near-extremal configuration admits, after removing $N^{o(1)}$ fraction of elements, a partition into $r^2 \approx N^{2/3}$ cells with the following constraints:

• No line meets more than $O(r)$ cells.
• No cell contains more than $O(N^{1/3})$ points.
• In most cells, only $O(N^{2/3})$ lines pass through the points.
• In each rich cell, $\Theta(|\text{cell}|^2)$ candidate lines exist, and a $\Theta(1)$ density appears in the actual incidence set. Proof techniques include random-sampling ("bush" argument) for balanced decomposition, the crossing-number inequality enforcing $L \approx M^2$ for lines and points per cell, and pruning mechanisms to maintain uniform bounds (Katz et al., 2023).

4. Success Criterion and Inverse Density

For generic choices of the $O(N^{1/3})$ parameters $(A,B,\ell_s)$, the construction succeeds on an $N^{-o(1)}$ fraction of its cells if $(L,P)$ is near-extremal:

• At least $N^{2/3-o(1)}$ cells are "good", each with $N^{1/3\pm o(1)}$ points.
• Each good cell produces $N^{2/3\pm o(1)}$ two-point lines.
• Summing over these cells, the aggregate number of incidences tracks $\sim N^{4/3-o(1)}$. Conversely, every successful parameter set yields an extremal arrangement, and every near-extremal arrangement is densely related to such constructions via projective changes. This pinpoints the universality and rigidity of the $\mathcal{T}_N$-recipe for extremal Szemerédi–Trotter examples (Katz et al., 2023).

5. Schematic Visualization and "Bush" Arguments

The construction's geometric essence is best conveyed graphically:

• The grid consists of vertical and horizontal cuts $(a_i, b_j)$.
• Structuring lines $\ell_1, \ldots, \ell_r$ cross the grid at generic slopes.
• In a rich cell, $N^{1/3}$ intersection points are marked; lines joining all these pairs are considered, with a subset highlighted if they cross $\geq N^{1/3}$ structuring lines in the cell.
• "Two-bush mixing" utilizes two far-separated "bush centers" in the projective plane, generating two families of parallel lines; their superposition induces the $r \times r$ cell structure. Generic far apart cells share $\gtrsim N^{1/3}$ lines, as substantiated by double-counting and crossing-number estimates.
• All extremal Szemerédi–Trotter arrangements, up to $N^{o(1)}$ errors and projective transformations, occur within this $O(N^{1/3})$-parameter family (Katz et al., 2023).

6. Analogous Cell Decompositions for Unit Circles

The methodology extends to analogous extremal arrangements involving unit circles instead of lines. The cell decomposition theorems and incidence bounds remain structurally parallel, affirming the utility and generality of $\mathcal{T}_N$-configurations in discrete geometry contexts (Katz et al., 2023).

7. Context, Implications, and Universality

The $\mathcal{T}_N$-configuration framework offers a uniform approach for realizing extremal incidence arrangements in both lines and other families, such as unit circles. After normalizing for projective equivalence and discarding negligible $N^{o(1)}$-fractional discrepancies, near-extremal configurations universally conform to or are densely approximated by successful instances of this recipe. This suggests a profound rigidity and centrality in the combinatorial geometry of extremal incidence arrangements (Katz et al., 2023).