Unbounded-degree number-field grids via infinite class-field towers (Golod–Shafarevich + point-counting)

Statement

The method. To beat a combinatorial extremal bound that is capped by the *fixed* arithmetic of a single quadratic ring — classically the Gaussian integers $\mathbb Z[i]$ underlying Erdős's $\sqrt n\times\sqrt n$ grid — replace $\mathbb Z[i]$ by the ring of integers $\mathcal O_K$ of a number field $K$ whose degree grows without bound while its arithmetic complexity (root discriminant) stays bounded. The unbounded degree supplies exponentially many algebraic integers of a *single* fixed absolute value (or a single fixed pattern of small sums/products); the bounded root discriminant keeps the Minkowski-embedded lattice inside a box of controlled size, so those integers pack into $\mathbb R^2$ (or $\mathbb R$) as a genuine finite point set. The two competing counts — how many points fit in the box vs. how many collisions of the target quantity they realize — no longer trade off through a factor that decays with the field, so a barrier of the form $n^{1+c/\log\log n}$ collapses to $n^{1+c}$ with $c>0$ absolute.

The infinite supply of such fields is exactly what infinite class-field towers provide: by Golod–Shafarevich a number field with enough tamely ramified primes has an infinite everywhere-unramified extension tower, hence fields of arbitrarily large degree with root discriminant *equal to that of the base* (unramified extensions do not change the root discriminant). The Hajir–Maire–Ramakrishna refinements make the tower *tame* and impose splitting conditions (a fixed prime $p$ splits completely, or infinitely many primes $q\equiv1\bmod4$ split), which is what lets one localize at a single archimedean/finite place. Ellenberg–Venkatesh geometry-of-numbers point counting does the final bookkeeping: an upper bound on lattice points in the box and a lower bound on the number of magnitude-$1$ (or few-value) elements.

Where it was first executed. This is the engine of the 2026 disproof of the Erdős unit-distance conjecture Erdős #90 — the unit distance conjecture (disproved 2026) (arXiv:2605.20695): a set of $n$ points in $\mathbb R^2$ with $\ge n^{1+c}$ unit distances, $c>0$ absolute, refuting the conjectured $n^{1+o(1)}$ upper bound. *(The construction was generated by an internal OpenAI model — per erdosproblems.com/90, last edited 20 May 2026 — and digested and human-verified by Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang, and Wood, arXiv:2605.20695; that is the one-line provenance, not the subject of this page. The mathematics below is the subject.)*

Where it already transferred. The same substitution — fixed quadratic ring $\to$ number field of degree $\asymp\log|A|$ — drives the 2026 disproof of the real sum-product conjecture Erdős #52 — the sum-product problem for integers (Bloom–Sawin–Schildkraut–Zhelezov, arXiv:2605.28781), the reason this is extracted as a portable concept rather than folded into either problem page.

Facts

- The old grid and its ceiling. Erdős (1946) took the $\sqrt n\times\sqrt n$ integer grid; the unit-distance count is governed by the number of representations of an integer as a sum of two squares, i.e. by the multiplicative structure of $\mathbb Z[i]$ (units $\{\pm1,\pm i\}$, unique factorization). This yields the lower bound $U(n)\ge n^{1+\Omega(1/\log\log n)}$ (erdosproblems.com/90; arXiv:2605.20695 §1). The $1/\log\log n$ decay is a fixed-ring artifact: the divisor-type function counting sum-of-two-squares representations is $n^{O(1/\log\log n)}$. Erdős conjectured this exponent shape is essentially optimal, i.e. $U(n)\le n^{1+o(1)}$, offering \$300 (in [Er82e]) resp. \$250 (in [Er83c],[Er85]) for a proof or disproof; the prior best *upper* bound is $O(n^{4/3})$ (Spencer–Szemerédi–Trotter [SST84]) (erdosproblems.com/90). - The new construction (arXiv:2605.20695, Thm 1.1). Replace $\mathbb Q(i)$ by CM fields $K_j=L_j(i)$, where $L_j$ is totally real of degree $f_j\to\infty$, all unramified outside a fixed finite set $T$ of odd primes and with a fixed prime $p$ completely split. Embed $\Lambda=p^{-2k}\mathcal O_{K_j}\hookrightarrow\mathbb C^f$ by the Minkowski map; take points from a bounded window $W=(a+\Lambda)\cap B_R$ and project to one complex place to land in $\mathbb R^2$. Result: there is an absolute $\varepsilon>0$ and point sets $\mathcal P_i\subset\mathbb R^2$ with $|\mathcal P_i|\to\infty$ and unit-distance count $\ge|\mathcal P_i|^{1+\varepsilon}$ — disproving $U(n)\le n^{1+o(1)}$. - Golod–Shafarevich, the infinite-tower engine (Golod–Shafarevich, "On the class field tower," Izv. Akad. Nauk SSSR 28 (1964) 261–272; statement per Wikipedia/MathWorld). For a pro-$p$ group $G$ with minimal generator number $d(G)$ and relator number $r(G)$: if $r(G)<d(G)^2/4$ then $G$ is infinite. Applied to a restricted-ramification Galois group $G_T^S$, arXiv:2605.20695 uses the explicit sufficient condition "if $r(G_T^S)\le d(G_T^S)^2/4$ then $G_T^S$ is infinite," giving an infinite tower of totally real fields with bounded root discriminant. Classic witnesses of the phenomenon are imaginary quadratic fields whose discriminant carries at least six prime factors (Wikipedia). - Bounded root discriminant = controlled arithmetic complexity. Because only the primes of $T$ ramify and only *tamely*, $|\mathrm{Disc}\,L|\le\prod_{p\in T}p^{[L:\mathbb Q]}$ (arXiv:2605.20695 §2), so the root discriminant $|\mathrm{Disc}\,L|^{1/[L:\mathbb Q]}\le\prod_{p\in T}p$ is bounded independent of degree. This is precisely what keeps the embedded lattice covolume-per-dimension bounded, so $\mathcal O_K$ fits many points into a fixed-radius box $B_R$. - Hajir–Maire–Ramakrishna supply tame towers with these splitting/discriminant guarantees: "Infinite class field towers of number fields of prime power discriminant," arXiv:1904.07062 (for every prime $p$, a solvable field ramified only at $p$ and $\infty$ with an infinite $p$-Hilbert class-field tower); "Cutting towers of number fields," Ann. math. Québec (2021) — the "cutting" method further lowers the root-discriminant bounds achievable in tame towers; and "On the Shafarevich group of restricted ramification extensions of number fields in the tame case," Indiana Univ. Math. J. 70 (2021). arXiv:2605.20695's abstract attributes the construction's ideas to Ellenberg–Venkatesh, Golod–Shafarevich, and Hajir–Maire–Ramakrishna. - Ellenberg–Venkatesh point counting ("The number of extensions of a number field with fixed degree and bounded discriminant," Annals of Math. 163(2) (2006) 723–741, arXiv:math/0309153; geometry of numbers). In arXiv:2605.20695 this is Lemma 2.1 (window count: $2\nu(\mathcal P)\ge(u\pi R^2/4v\delta^2)^f$ while $|\mathcal P|\le(9R^2/\delta^2)^f$) and Lemma 2.2 (a lower bound $|U|\ge\prod_{j=1}^s(k_j+1)/h(K)$ on the magnitude-$1$ elements $u$ with $|u|=1$ in every embedding, $s$ = number of split primes above $p$, $h(K)$ = class number). The ratio of these two counts is what produces the *absolute* exponent gain. - Explicit exponent (arXiv:2605.20695). With $T=\{3,5,7,11,13,17\}$ and $p=101$ the argument certifies $\varepsilon\approx6.24\cdot10^{-38}$ — tiny, but a fixed positive constant, which is all that is needed to break $n^{1+o(1)}$. - Migration to sum-product (arXiv:2605.28781). Bloom–Sawin–Schildkraut–Zhelezov build arbitrarily large $A\subset\mathbb R$ whose elements are algebraic integers in a number field of degree $\asymp\log|A|$ with $\max(|A+A|,|AA|)\le|A|^{2-c}$, $c>0$ absolute — disproving the real sum-product conjecture Erdős #52 — the sum-product problem for integers. Same paper: the many-sums-and-products conjecture fails, $\max(|kA|,|A^{(k)}|)\le|A|^{C\log k/\log\log k}$ for every $k\ge3$; and analogous constructions hold for $p$-adics, finite fields, and function fields in positive characteristic. Notably the degree needed here is only $\asymp\log|A|$ — markedly less number-theoretic machinery than the unit-distance construction — the sum-product target is softer. - Inspiration link (documented). Per this wiki's problems/52 and the BSSZ26 introduction, the authors state they "were inspired to revisit the possibility of disproving the sum-product conjecture using number fields of large degree by the recent OpenAI counterexample to the unit distance conjecture" — i.e. Erdős #90 — the unit distance conjecture (disproved 2026) $\to$ Erdős #52 — the sum-product problem for integers technique transfer through this exact machinery.

Technique

Recombination-ready recipe — when a combinatorial extremal count is stuck at $n^{1+o(1)}$ because the natural construction lives in one fixed quadratic (or cyclotomic) ring:

1. Diagnose the barrier as a fixed-ring artifact. The tell-tale is an exponent $1+c/\log\log n$ (or a bound tied to a divisor-type / representation function) coming from a *single* ring like $\mathbb Z[i]$: units and factorization there cap the multiplicity of your target relation. If the cap scales down with something that itself grows only because the ring is fixed, degree is the free parameter you are not using. 2. Re-express the target as counting algebraic integers with a fixed invariant. Unit distances $\Leftrightarrow$ algebraic integers of a single absolute value $1$ (magnitude-$1$ elements of a CM field, i.e. the "unit circle" over the totally real subfield). Small sumset/productset $\Leftrightarrow$ elements confined to a low-complexity additive-and-multiplicative substructure. The relation must be one whose *number of realizations grows with the degree of the field*. 3. Choose a family of number fields with degree $\to\infty$ and root discriminant bounded. This is not optional decoration — it is the whole point. Bounded root discriminant $\Rightarrow$ the Minkowski lattice $\mathcal O_K\subset\mathbb R^{[K:\mathbb Q]}$ has bounded covolume per dimension $\Rightarrow$ a fixed-radius box holds a controlled number of points (finite construction). Get the family from an infinite class-field tower: verify the Golod–Shafarevich inequality $r<d^2/4$ (enough tamely ramified primes), then pass up the (unramified, hence root-discriminant-preserving) tower for unbounded degree. Use Hajir–Maire–Ramakrishna tame towers when you also need a splitting condition (a chosen prime splits completely; infinitely many $q\equiv1\bmod4$ split) to localize at one place. 4. Localize to $\mathbb R^2$ or $\mathbb R$ at a single place. Project $\mathcal O_K$ (or a coset $a+p^{-2k}\mathcal O_K$) through one complex embedding into a bounded window $B_R$; the completely-split prime $p$ guarantees the local structure you exploited survives the projection. 5. Two-sided count (Ellenberg–Venkatesh geometry of numbers). Upper-bound $|\mathcal P|$ = lattice points in the window (a volume count), and lower-bound the number of target-collisions = magnitude-$1$ / few-value elements (a count over split primes, $\ge\prod(k_j+1)/h(K)$). Their ratio yields the exponent; because both counts scale the same way in the degree while the *collision multiplicity* does not decay, the surplus is an absolute $\varepsilon>0$, no longer $o(1)$. 6. Migrate. The construction is agnostic to *which* extremal quantity you froze in step 2. Having broken unit distances (frozen absolute value), swapping the frozen invariant to "small additive + multiplicative footprint" reuses the identical tower/point-count skeleton to break sum-product — with a *weaker* field requirement (degree $\asymp\log|A|$ suffices, arXiv:2605.28781). The tower machinery is the reusable core; the frozen invariant is the interchangeable part.

WHY it works (the mechanism, for recombination). A fixed ring like $\mathbb Z[i]$ has a fixed unit group and a fixed factorization type, so any quantity you build from "how many elements share an invariant" is capped by that ring's arithmetic — and the cap manifests as the $\log\log n$ decay. Number-field *degree* is a second, unbounded axis of arithmetic freedom: higher degree $\Rightarrow$ more magnitude-$1$ units, more representations, richer collision structure. Naively raising the degree would blow up the discriminant and disperse the lattice, destroying the finite construction; the bounded-root-discriminant tower is exactly the device that raises degree *for free*, because unramified extensions preserve the root discriminant. Golod–Shafarevich guarantees the tower is infinite (degree unbounded); tameness + splitting (Hajir–Maire–Ramakrishna) makes it usable at one place; geometry-of-numbers point counting (Ellenberg–Venkatesh) converts "abundant magnitude-$1$ integers in a bounded box" into an explicit combinatorial surplus. The method is powerful precisely because the hard input — an infinite tower with controlled complexity — is a *reusable black box from algebraic number theory*, and the problem-specific work reduces to identifying which invariant to freeze and checking one point-count inequality.

Related

- Erdős #90 — the unit distance conjecture (disproved 2026) — the unit-distance problem; its 2026 disproof ($U(n)\ge n^{1+c}$, absolute $c>0$) is the first execution of this method, replacing Erdős's $\mathbb Z[i]$ grid by CM-field towers (arXiv:2605.20695). - Erdős #52 — the sum-product problem for integers — the sum-product problem; the *real* case was disproved by the same substitution (fixed ring $\to$ degree-$\asymp\log|A|$ number field, arXiv:2605.28781), the documented migration that justifies extracting this as a concept. The integer case #52 stays open — plain $\mathbb Z$ has none of the class-field-tower / magnitude-$1$-unit structure this method needs. - Elekes–Sharir(–Guth–Katz) reduction: distinct distances → point-line incidences in SE(2) — the parallel "extract the transferable method into a concept page" case in this wiki (pulled out of the distinct-distances problems); like this page, its power is a low-complexity algebraic reduction feeding a counting bound. - Incidence geometry: Szemerédi–Trotter theorem, the crossing lemma, and Zarankiewicz-type bounds — the classical route to unit-distance *upper* bounds (Spencer–Szemerédi–Trotter $O(n^{4/3})$); this method attacks the *lower* bound / construction side that incidence geometry cannot reach. - Finite-field / projective-plane constructions for extremal additive sets — arXiv:2605.28781 also realizes the construction over finite fields and function fields; the tower machinery is characteristic-agnostic. - Lean 4 formalization of constructions and conditional reductions (Erdős-problem context) — natural target: formalize which step of the number-field construction *fails* over $\mathbb Z$ itself, sharpening the open status of the integer case of #52.