On the number of perfect positive definite quadratic forms

Abstract: We prove a new asymptotic lower bound for the number of perfect positive definite quadratic forms in n variables which is close to the optimal one.

• The paper refines the asymptotic lower bound for perfect quadratic forms, proving Pₙ ≥ exp((n² log n)/2 - O(n²)).
• It employs explicit constructions using Aₙ root lattices and combinatorial group theory to count inequivalent positive definite quadratic forms.
• The refined results bridge the gap with classical upper bounds, advancing benchmarks for lattice sphere packings and discrete geometry.

Asymptotic Lower Bound for Perfect Positive Definite Quadratic Forms

Introduction and Context

The paper "On the number of perfect positive definite quadratic forms" (2607.04239) provides a significant refinement of the known asymptotic lower bounds for the number of perfect, positive definite quadratic forms in $n$ variables, denoted $P_n$. Perfect forms, as introduced by Voronoi, are central objects in the theory of quadratic forms and lattice sphere packings, with direct implications for discrete geometry and number theory.

Background results establish that, for each $n$, there are only finitely many $P_n$ up to equivalence, and explicit values are known for small $n$. However, for $n \gg 1$, only exponential upper and lower bounds were previously available, with the best known lower bound being $P_n > \exp(n^{1-\varepsilon})$ for any $\varepsilon > 0$ (Bacher, 2018), whereas the classical upper bound is $P_n < \exp(O(n^2 \log n))$ (van Woerden, 2020).

Main Results and Approach

The main result of the paper is a new, asymptotic lower bound:

$$P_n \geq \exp\left(\frac{n^2 \log n}{2} - O(n^2)\right)$$

as $P_n$0, which nearly matches the upper bound at the exponential scale and therefore narrows the asymptotic uncertainty by orders of magnitude compared to previous results.

The crux of the construction relies on the properties of the root lattices $P_n$1, their symmetry, and the arithmetic and geometric structure of sublattices of prime index. The author constructs families of lattices $P_n$2, indexed by integer vectors $P_n$3 with entries in $P_n$4, and shows under suitable conditions that the corresponding quadratic forms are perfect and pairwise inequivalent up to isometry.

A careful combinatorial and group-theoretic analysis ensures that most choices of $P_n$5 yield new forms, and the arguments employ tools from the geometry of numbers (in particular, analysis of lattice points in convex bodies), Abelian group structure of factor groups, and properties of quadratic forms associated to these lattices.

Technical Contributions

• Explicit Construction: For each $P_n$6 and suitable prime $P_n$7, a family of sublattices $P_n$8 of index $P_n$9 in $n$0 is constructed, parametrized by $n$1 choices of $n$2.
• Group Structure and Equivalence Classes: The analysis of the quotient groups $n$3 determines when two forms are equivalent and counts the number of distinct forms.
• Extremality and Perfection: Using Voronoi’s and Baranovskii’s criteria, the paper shows that for most constructed lattices, the corresponding quadratic forms are extremal and thus perfect.
• Avoidance via Probabilistic Arguments: Employing a variant of the Minkowski-Hlawka theorem's mean-value argument, the construction avoids unwanted overlaps in the support of the associated lattices, ensuring a large number of inequivalent perfect forms.
• Sharp Cardinality Estimates: For well-chosen $n$4 and after discarding degenerate cases, the cardinality of the resulting perfect forms matches the upper bound up to a sub-exponential factor, i.e., within the exponential term in $n$5.

Strong Numerical and Structural Claims

• Sub-exponential Gap: The lower bound established is exponentially stronger than previously known constructions, moving from $n$6 to essentially $n$7.
• Uniqueness of Certain Subgroups: The argument leverages the uniqueness of subgroups of order $n$8 in associated factor groups under certain congruence conditions, ensuring the constructed forms are mostly inequivalent, up to a factor $n$9 due to isometries of $P_n$0.
• Asymptotic Optimality: The upper and lower bounds now match up to $P_n$1 in the exponent, suggesting that the true asymptotics of $P_n$2 are now tightly constrained.

Implications and Future Directions

The result represents a decisive advancement in understanding the asymptotics of perfect quadratic forms, which are tightly linked to the fundamental limits of lattice sphere packings, classifications of high-dimensional lattices, and various problems in discrete optimization and coding theory. Theoretically, it suggests that the structure of perfect forms in high dimensions is extremely rich, almost saturating the combinatorial possibilities permitted by the upper bound.

Practically, the refined lower bound enables improved benchmarks and expectations for algorithms in lattice enumeration, sphere packing, and related cryptographic constructions. From an algorithmic perspective, the explicitness of the construction offers pathways for generating large numbers of inequivalent perfect forms suitable for computational experiments in high dimensions.

Potential future directions include:

• Sharper Upper Bounds: Whether the $P_n$3 exponent in the upper bound is the precise threshold remains unresolved; a matching constant factor remains an open problem.
• Structure of Extremal and Perfect Forms: The qualitative structure and classification of the constructed perfect forms merit further investigation, particularly their geometric and automorphic properties.
• Extensions to Other Lattice Families: The techniques could potentially be generalized to other classes of lattices, possibly yielding even denser constructions in certain dimensions or analogous results for other types of forms.

Conclusion

By constructing a family of extremal (and hence perfect) positive definite quadratic forms whose count matches the classical upper bound up to sub-exponential factors, the paper (2607.04239) substantially clarifies the asymptotic landscape of $P_n$4. The results close a longstanding gap between upper and lower bounds and offer new avenues for both theoretical and computational developments in the geometry of numbers, quadratic forms, and related applications.