The Minkowski grid has robustly many repeated distances

Abstract: We show that there exists a constant $δ&gt; 0$ such that for any positive integer $n$ there exists a set of $n$ points $P \subset \mathbb{R}<sup>2$ with the following property: for every subset $A \subseteq P$ of size $|A| \geq 2$, [ \max_{λ>0} #{(a,b)\in A \times A: a\ne b,\ \lvert a-b\rvert=λ} \gtrsim \frac{|A|^2}{n^{1-δ}}.] Our result is a vertical amplification of a robust Ramanujan estimate recently established by Croot-Mao-Pohoata-Sheffer-Yip for arbitrary subsets of the ordinary square grid, and is inspired by recent constructions for the Erdős unit distance problem and the Elekes-Rónyai problem. Taking $A=P$, the inequality above gives a distance occurring $n<sup>{1+δ}$ times in $P$; thereby a scaled copy of $P$ is a counterexample for the unit-distance conjecture. In addition, the same inequality shows that (1) all subsets of $P$ of size $\gtrsim n<sup>{1-δ}$ must contain isosceles triangles, and (2) all subsets of $P$ of size $\gtrsim n<sup>{1/2-δ}$ must contain repeated distances. These features give polynomially improved estimates for old problems of Erdős. The existence of a set satisfying property (1) confirms a conjecture of Erdős from 1980, whereas the existence of a set with property (2) answers a question of Conlon-Fox-Gasarch-Harris-Ulrich-Zbarsky in the negative.

• The paper establishes an explicit construction using high-degree totally real number fields to create an n-element set with robust repeated distances.
• The paper employs Minkowski embedding and combinatorial sieve techniques to derive a lower bound of μ(A) ≳ |A|²/n^(1-δ) for every sufficiently large subset.
• The paper shows that any large subset of the grid must contain isosceles triangles or repeated distances, resolving longstanding conjectures in discrete geometry.

The Minkowski Grid and Robust Multiplicity of Distances

Overview

The paper “The Minkowski grid has robustly many repeated distances” (2607.05374) establishes that, for each $n$, there exists an explicit $n$-element subset $P \subset \mathbb{R}^2$—derived from the ring of integers of a high-degree totally real number field—such that every sufficiently large subset $A \subseteq P$ necessarily contains many repeated distances. Quantitatively, for some absolute $\delta > 0$, every $A \subseteq P$ of size $|A| \geq 2$ satisfies

$\mu(A) \gtrsim \frac{|A|^2}{n^{1-\delta}},$

where $\mu(A)$ denotes the maximum number of (ordered) pairs at any fixed distance in $A$. When $n$0, this construction yields a distance that appears $n$1 times, contradicting the classical Erdős unit distance conjecture in a robust manner. The results give substantial new bounds for related extremal configurations in discrete geometry and address longstanding open problems on configurations lacking isosceles triangles or repeated distances.

Context and Prior Work

The classical Erdős unit distance conjecture posited that a set of $n$2 points in the plane determines at most $n$3 unit distances for any $n$4, for some constant $n$5. A classical construction using integer grids and number-theoretic results (notably, the Ramanujan estimate on $n$6) already produces distances repeating superlinearly, though only by a thin margin.

Recent work, including that of Croot, Mao, Pohoata, Sheffer, and Yip (CMPSY), introduced combinatorial sieve techniques yielding “robust Ramanujan-type” lower bounds for the multiplicity of distances within subsets of grid structures. Separately, constructions leveraging the arithmetic of number fields (notably by OpenAI and collaborators) have produced explicit counterexamples to the original Erdős conjecture; these techniques utilize “vertical amplification” by considering multidimensional lattices from number fields where many primes split completely.

The present work unites and extends these lines: it gives constructions that not only create many repeated distances in the full set, but guarantee robust multiplicity of distances for all sufficiently large subsets, yielding new bounds for extremal subconfigurations avoiding isosceles triangles or repeated distances.

Main Results and Techniques

Robust Lower Bound on Distance Multiplicity

The principal theorem asserts the following: there exists $n$7 such that for any positive integer $n$8, there is a set $n$9 of size $P \subset \mathbb{R}^2$0 such that for all subsets $P \subset \mathbb{R}^2$1 with $P \subset \mathbb{R}^2$2,

$P \subset \mathbb{R}^2$3

implying some fixed distance occurs at least $P \subset \mathbb{R}^2$4 times in $P \subset \mathbb{R}^2$5. This generalizes and strengthens the CMPSY robust sieve bound, which previously required subtle pigeonhole and exponential factors in $P \subset \mathbb{R}^2$6.

Extremal Subset Results

Two corollaries are central:

1. Isosceles triangle avoidance: Any subset $P \subset \mathbb{R}^2$7 with $P \subset \mathbb{R}^2$8 contains an isosceles triangle. This closes a conjecture of Erdős (1980).
2. Distinct distance subsets: Any subset $P \subset \mathbb{R}^2$9 with $A \subseteq P$0 contains a repeated distance. This resolves a problem posed by Conlon, Fox, Gasarch, Harris, Ulrich, and Zbarsky, showing that $A \subseteq P$1 is not possible in the worst case.

These results represent polynomial improvements over previous upper bounds, moving beyond exponential factors in log terms.

Construction and Proof Outline

The configurations are constructed as follows:

• Number Field Construction: A tower of totally real fields $A \subseteq P$2 of bounded root discriminant and doubling degree is selected (using results of Hajir-Maire-Ramakrishna). Infinitely many primes split completely in every $A \subseteq P$3, enabling uniformity in local conditions.
• Minkowski Embedding: The set $A \subseteq P$4 is taken as an $A \subseteq P$5-element subset of $A \subseteq P$6, where $A \subseteq P$7 is a box in the ring of integers $A \subseteq P$8 embedded into $A \subseteq P$9 via the Minkowski map.
• Combinatorial Sieve: For many split primes, the authors define lattices $\delta > 0$0 of pairs differing by prescribed local congruence classes mod each split place. The Cauchy-Schwarz method, amplified over all sign choices, guarantees that differences in $\delta > 0$1 cover the “good” classes robustly.
• Parameter Optimization: By taking the field degree and number of split primes sufficiently large, and controlling the discriminant growth, the construction achieves the desired polynomial improvement in bounds.

The arguments exploit the independence of local splitting conditions, the sharpness of Minkowski box lattice counts, and Ramanujan-type enhancements for sieve bounds.

Implications and Significance

Theoretical Consequences

• Disproving the Unit Distance Conjecture: The existence of explicit planar sets with distance multiplicities polynomially above $\delta > 0$2 definitively settles the unit distance problem for general sets.
• Extremal Subset Size Compression: Previous expectations based on analogies to arithmetic progressions and Sidon sets (for subsets of $\delta > 0$3) suggested weaker upper bounds. The results show genuine polynomial separation between the subset problems for the plane versus those for one-dimensional analogues.
• Link to Higher-Order Geometric Hypergraphs: The proof techniques connect to the hypergraph container method of Balogh and Solymosi (for collinearity). Here, metric conditions replace combinatorial or linear ones, opening a new front for hypergraph extremal questions in geometric combinatorics.

Methodological Advances

• Vertical Amplification: The use of high-degree number fields and Minkowski lattices enables simultaneous “amplification” of sieve effects across dimensions, outstripping what is possible via purely combinatorial or classical additive tools.
• Unified Construction: The configuration suffices not just for unit distance counterexample, but for bounding independence numbers of extremal hypergraphs defined by isosceles triangles or repeated distances, with a single explicit set.

Speculation on Future Directions

• Further Generalizations: The principles underlying the vertical amplification and robust sieve mechanism are likely extendable to other geometric and combinatorial incidence problems, especially for structures defined via local-global or arithmetic constraints.
• Quantitative Optimization: While the construction is robust, optimizing the actual value of $\delta > 0$4, as well as tightening constants, remains open. Improvements in lattice point counting and sieve loss analyses may yield even sharper exponents.
• Beyond the Plane: Extensions to higher dimensions, or to other normed spaces, could yield insight into analogous questions in $\delta > 0$5, potentially illuminating long-standing conjectures about repeated distances or extremal subset sizes in those settings.
• Connections to Additive Combinatorics: The connections of these geometric questions to classical problems on Sidon sets, sum-product estimates, and arithmetic progressions suggest further fruitful cross-pollination between discrete geometry and additive number theory.

Conclusion

This work provides a definitive, quantitatively strong construction of planar point sets for which repeated distances, isosceles triangles, and subset extremal phenomena are all constrained in a polynomially robust manner. The technical toolset, employing constructions from algebraic number theory and sophisticated sieve techniques, advances the state of knowledge in extremal discrete geometry, addresses classical conjectures, and likely charts a path for future progress on related geometric and arithmetic problems in combinatorics.