Asymptotic Classification of Diophantine Equilibrium in the Base {2, 3, 5}: Lattice Geometry, Harmonic Proof, and Explicit Residues

1. Setup and Definitions

Let

$$\mathcal{S}\left(n\right)=\{(a,b,c)\in {\mathbb{Z}}_{\ge 0}^3:2a+3b+5c=n\}.$$

For $s=(a,b,c)$ define

$$\sigma \left(s\right)=\sqrt{{\displaystyle \frac{{(a-\overline{x})}^2+{(b-\overline{x})}^2+{(c-\overline{x})}^2}{3}}},\overline{x}={\displaystyle \frac{a+b+c}{3}}.$$

Equivalently, minimize the quadratic form

$$Q(a,b,c)=3(a^2+b^2+c^2)-{(a+b+c)}^2=9{\sigma }^2.$$

(1)

Let $\mathcal{M}\left(n\right)\subset \mathcal{S}\left(n\right)$ be the set of minimizers of Q, and $m\left(n\right)=\left|\mathcal{M}\right(n\left)\right|$.

2. Geometric Framework: Lattices and Voronoi

Let $g=(2,3,5)$ and $H_n=\{x\in {\mathbb{R}}^3:g·x=n\}$. The form

$$Q\left(x\right)=x^{\top }Bx,B=3I-\mathbf{1}{\mathbf{1}}^{\top },$$

is strictly convex on $H_n$ [1][Chap. 1]. Lagrange multipliers give the unique continuous minimizer

$$x^{\left(n\right)=\left(\frac{n}{10},\frac{n}{10},\frac{n}{10}\right)}.$$

Fix an integral particular solution $s_0\left(n\right)$ to $g·x=n$, and set

$$K=\{z\in {\mathbb{Z}}^3:g·z=0\},$$

the homogeneous lattice (rank 2). A convenient basis is

$$u=(1,1,-1),v=(-4,1,1),$$

(2)

since $2+3-5=0$ and $-8+3+5=0$. Every $x\in \mathcal{S}\left(n\right)$ can be written as $x=s_0\left(n\right)+z$ with $z\in K$.

Lemma 1 

(Reduction to CVP). Let $y\left(n\right)=x^{\left(n\right)-s_0\left(n\right)}$. Then

$$\mathcal{M}\left(n\right)=arg\underset{z\in K}{min}Q\left(z-y\left(n\right)\right),m\left(n\right)=\#arg\underset{z\in K}{min}Q\left(z-y\left(n\right)\right).$$

Thus $m\left(n\right)$ equals the number of nearest neighbors to $y\left(n\right)$ in K under the metric Q (Voronoi cells) [2,3] [Ch. 21].

Lemma 2 

(Ties ⇒ hyperplanes linear in n). For $z_1,z_2\in K$, the condition $Q(z_1-y\left(n\right))=Q(z_2-y\left(n\right))$ reduces to

$$2{\langle z_1-z_2,y\left(n\right)\rangle }_Q=Q\left(z_1\right)-Q\left(z_2\right),$$

where ${\langle u,v\rangle }_Q=u^{\top }Bv$. Since $y\left(n\right)$ is affine in n for large n, tie values form arithmetic progressions (Voronoi walls) [3,4].

Lemma 3 

(Finite periodicity). There is a period T with $T\mid 30$ such that, for large n, the tie pattern (hence $m\left(n\right)$) is T-periodic in n (nonnegativity affects only finitely many n) [7][Chap. V].

Theorem 1 

(Asymptotic classification). There exist $N_0$ and a finite set $\mathcal{C}\subset \{0,1,\dots ,T-1\}$ with $T\mid 30$ such that, for all $n\ge N_0$,

$$m\left(n\right)=\left\{\begin{array}{cc}2,\hfill & nmodT\in \mathcal{C},\hfill \\ 1,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

In particular, $m\left(n\right)\in \{1,2\}$ asymptotically.

Proof. 

Combine Lemmas 1, 2, 3, and the fact that in rank 2 generic ties are binary (Voronoi walls between two cells) [3].    □

3. Harmonic Proof: Lattice Theta and Poisson

For $t>0$ and $y\in {\mathbb{R}}^3$ define the theta series

$${\mathrm{\Theta }}_K(t,y)=\sum _{z\in K}exp\left(-\pi tQ(z-y)\right).$$

For each n, take $y\left(n\right)=x^{\left(n\right)-s_0\left(n\right)}$. A Gibbs concentration yields:

Lemma 4 

(Concentration). Let $Q_{min}\left(n\right)={min}_{z\in K}Q(z-y\left(n\right))$ and

$$S_t\left(n\right)=\sum _{z\in K}exp\left(-\pi t(Q(z-y\left(n\right))-Q_{min}\left(n\right))\right).$$

Then ${lim}_{t\to \infty }{S}_t\left(n\right)=m\left(n\right)$, with uniform convergence off walls.

Let $K^{\#}=\{w:{\langle w,z\rangle }_Q\in \mathbb{Z}\forall z\in K\}$ be the dual lattice under Q. The Poisson summation formula (see [1][Sec. VII.2], [5][Ch. 1], [6][Ch. 4]) gives

$${\mathrm{\Theta }}_K(t,y)={\displaystyle \frac{1}{\mathrm{covol}\left(K\right)}}{t}^{-1}\sum _{w\in K^{\#}}exp\left(-{\displaystyle \frac{\pi }{t}}{\parallel w\parallel }_Q^2\right)e^{2\pi i{\langle w,y\rangle }_Q},$$

with ${\parallel w\parallel }_Q^2={\langle w,w\rangle }_Q$.

Lemma 5 

(Exact periodicity of phases). Write $y\left(n\right)=\alpha n-\beta \left(n\right)$ with $\alpha =\frac{1}{10}\mathbf{1}$ and $\beta \left(n\right)=s_0\left(n\right)$ (integral, 30-periodic). Then

$$e^{2\pi i{\langle w,y\left(n\right)\rangle }_Q}=e^{2\pi in{\langle w,\alpha \rangle }_Q}·e^{-2\pi i{\langle w,\beta \left(n\right)\rangle }_Q}.$$

As α has denominator 10 and $\beta \left(n\right)$ is 30-periodic and integral, the phases are roots of unity and $n\mapsto {\mathrm{\Theta }}_K(t,y\left(n\right))$ is exactly T-periodic for some $T\mid 30$.

Theorem 2 

(Harmonic version of Theorem 1). For each $t>0$, $n\mapsto {\mathrm{\Theta }}_K(t,y\left(n\right))$ is T-periodic with $T\mid 30$ and has a finite Fourier series over $\mathbb{Z}/T\mathbb{Z}$. Taking $t\to \infty$ in $S_t\left(n\right)$ of Lemma 4, we obtain that $n\mapsto m\left(n\right)$ is T-periodic for $n\ge N_0$ and $m\left(n\right)\in \{1,2\}$.

Remark 1 

(Discrete spectrum). The spectrum of $n\mapsto m\left(n\right)$ is contained in rational frequencies $\{j/T\}$ (Fourier–Bohr) [5][Ch. 1].

4. Explicit Residues with $m\left(n\right)=2$ (Evidence)

Exact computations indicate that, for sufficiently large n, ties occur precisely on the classes

$$\begin{array}{|c|}\hline \mathcal{C}=\{4,5,6,14,15,16,24,25,26\}(mod30)\\ \hline\end{array}.$$

Computational fact. For $N_0=1000$ and all n with $1000\le n\le 7000$:

$$m\left(n\right)=\left\{\begin{array}{cc}2,\hfill & \mathrm{if}nmod30\in \mathcal{C},\hfill \\ 1,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

No instance with $m\left(n\right)>2$ was observed. This matches the Voronoi picture (binary ties in rank 2) and the harmonic periodicity (Section 3). The refinement modulo 30 reflects that $x^{\left(n\right)=(n/10)\mathbf{1}}$ and parities/periodicities of $s_0\left(n\right)$ depend on denominators dividing 10 and on $\mathrm{lcm}(2,3,5)=30$.

5. Arithmetic Observations

Observation 5.1 

(Primes and external prime powers). In the verified ranges, for every prime $p\ge 7$ we have $m\left(p\right)=1$, and for every prime power $q^k$ with $q\notin \{2,3,5\}$ we have $m\left(q^k\right)=1$. Theorem 1 explains that, up to finitely many small exceptions, this stabilizes for $n\ge N_0$.

Remark 2 

(On prior conjectures). Lower bounds of the form $m\left(n\right)\ge \left({\displaystyle \genfrac{}{}{0pt}{}{k}{2}}\right)$ (with k the number of distinct prime factors of n) are not correct in general: the asymptotic structure is governed by modular periodicity (Voronoi/Poisson), not by factorization alone.