propositionnovelstructure_uncertain

No algebraic configuration with large heights

Proposition·alg:obstruction·§8 The algebraic obstruction

Statement (paper)

There is no sequence with the following properties. At index nn, let tn→∞t_n\to\infty, let Jn→∞J_n\to\infty, and let Pn⊂R2P_n\subset\R^2 be a finite set of algebraic points with ∣Pn∣≤tn1+o(1)\lvert P_n\rvert\le t_n^{1+o(1)}. Let Gn⊂Pn×PnG_n\subset P_n\times P_n be a bipartite graph, using two copies of PnP_n, with at least tn2−o(1)t_n^{2-o(1)} edges. To each edge (p,r)(p,r) assign a nonzero algebraic number λpr\lambda_{pr}. Suppose that, for an absolute c>0c>0, (zp−zr)(wp−wr)=2−λpr−λpr−1,h(λpr)≥cJn,h(λprλp′r′λpr′λp′r)=o(Jn)\begin{aligned} (z_p-z_r)(w_p-w_r) &=2-\lambda_{pr}-\lambda_{pr}^{-1}, \\ h(\lambda_{pr})&\ge cJ_n, \\ h\left(\frac{\lambda_{pr}\lambda_{p'r'}} {\lambda_{pr'}\lambda_{p'r}}\right)&=o(J_n) \end{aligned} uniformly over edges and complete rectangles, respectively. Finally, suppose that no edge has both endpoints on a unit circle containing at least tn9/10t_n^{9/10} points of PnP_n. Here hh is the absolute height from .

In plain English

No sequence exists of finite algebraic point sets PnP_n of size tn1+o(1)t_n^{1+o(1)} with a bipartite graph of tn2−o(1)t_n^{2-o(1)} edges, edge numbers λpr\lambda_{pr} satisfying (zp−zr)(wp−wr)=2−λpr−λpr−1(z_p-z_r)(w_p-w_r)=2-\lambda_{pr}-\lambda_{pr}^{-1}, heights h(λpr)≥cJnh(\lambda_{pr})\ge cJ_n, rectangle cross-ratios of height o(Jn)o(J_n), and no edge inside a unit circle with at least tn9/10t_n^{9/10} points.

Why it's here

This is the algebraic obstruction that rules out the extremal scenario of many unit distances with large-height ratios, and it is what the main theorem's power saving rests on. The proof assumes such a sequence and finds a dense grid whose ratios factor into row, column and constant parts. Differentiating the distance equations gives a linear system with radical coefficients, and a Galois sign flip of an unused radical contradicts a nonzero derivative.}
Classified novel:
This is the paper's central new algebraic obstruction ruling out the large-height extremal configuration, on which the power saving below the Szemerédi–Trotter n4/3n^{4/3} bound rests. No prior unit-distance argument (e.g. Spencer–Szemerédi–Trotter, crossing-number methods) uses height constraints on edge numbers with a Galois sign-flip argument.

Uses

Nothing else in the paper

Used by

thmPower-saving bound for planar unit distances

Inside the proof

claimEdge ratios transcendental, rectangle defects constantclaimMinimal loci are symmetric, not a common unit circleclaimGeneric points embed function fieldsclaimGrid ratios factor through constant defectsclaimDifferentiated grid system with radicalsclaimUnused sign change contradicts nonzero derivative

Sources

PaperOpen
Section
§8 The algebraic obstruction
Page
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Label
alg:obstruction
Line
sections/07-algebra.tex:14
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    structure_uncertain
    No proof environment; proved by the running text of Section 8 (lines 40-678), assembling Lemmas 8.2-8.6, ending with 'This proves Proposition alg:obstruction'.