A Fourier Consequence for Lattice Kissing Numbers and Average Contacts

For a rank-$n$ Euclidean lattice $L$, we write $\min L=\min_{0\ne x\in L}\langle x,x\rangle$ for the minimum squared length of nonzero vectors in $L$; the kissing number

counts the number of such vectors, including both signs. The maximal lattice kissing number in dimension $n$ is

In this note, we obtain a new asymptotic upper bound for $K_{\mathrm{lat}}(n)$, using explicit auxiliary functions constructed in OpenAI’s preprint [15].

Theorem 1. As $n\to\infty$,

Here $\frac{1}{2}\log_{2}(2e/\pi)=0.395599455708322\ldots$.

For a packing of congruent balls, the contact degree of a ball is the number of other balls tangent to it; the average contact degree of a nonempty finite packing is the arithmetic mean of these degrees. After scaling such balls to radius $1/2$, their centers form a finite set $C\subset\mathbb{R}^{n}$ with minimum separation at least 1, and the average contact degree is

Corollary 4 gives the same upper bound as Theorem 1 for $\overline{\tau}(C)$, uniformly over all such $C$.

To place the bound in context, let $K(n)$ be the ordinary kissing number—i.e., the largest number of nonoverlapping unit balls that can touch one unit ball in $\mathbb{R}^{n}$. Equivalently, $K(n)$ is the largest size of a set of unit vectors with pairwise inner products at most $1/2$. Normalizing the shortest vectors of a lattice gives such a set, so $K_{\mathrm{lat}}(n) \le K(n)$. The classical spherical-code bound of Kabatiansky and Levenshtein [12], from 1978, gives the exponent $0.40141\ldots$ for $K(n)$. Combining it with Sidel’nikov’s spherical-cap reduction [18] gives the optimized classical bound [12], with exponent $0.400944\ldots$; subsequent work of Sardari and Zargar [17], and of Zargar [22], improved multiplicative factors.

Fourier methods offer a second route, through the contacts in a packing. In 2020, Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini [1] extrapolated an implied kissing exponent of $0.3956$ (the $d=\infty$ row of [1]) from their numerically optimized density bound. Its interpretation as an average-contact bound relied on interpolation and sign conjectures. Their Theorem 5.3 gives an unconditional inequality for admissible auxiliary functions; we apply this result to the explicit functions of [15], obtaining an exponent that agrees with their extrapolation to the displayed precision, without relying on the intermediate conjectures.

The cap-optimized spherical-code hierarchy of [15] gives

at level 2; see [15]. Our Theorem 1 and Corollary 4 give a smaller exponent for lattice kissing numbers and average contact degree, respectively. In fact, Lemma A.1 in Appendix A shows that the infimum of the cap-optimized fixed-level asymptotic kissing exponents of the hierarchy in [15] remains strictly above ours. To our knowledge, Theorem 1 gives the first proven asymptotic upper bound for lattice kissing numbers whose exponential rate is strictly smaller than the best known upper-bound rate for the unrestricted kissing number.²

Meanwhile, there remains a large gap between the upper and lower bounds for lattice kissing numbers. The Barnes–Wall lattices [3] in dimensions $n=2^{m}$ have kissing numbers

see Nebe [14]. Recent work of Laarhoven and the present author [13] improves this to

as $n\to\infty$ through prime powers.³

Our analytic input comes from Chapter 1 of [15]; all references to that preprint use the version updated August 6, 2026. That chapter determines the exponential strength of the Fourier linear program for sphere-packing density, developed by Gorbachev [11] and Cohn–Elkies [6]. Writing $\mathrm{LP}_{n}$ for the infimum of the density upper bounds supplied by this program, [15] states

Consequently, the maximal packing density $\Delta_{n}$ satisfies $\Delta_{n}\le2^{-(0.604400544\ldots+o(1))n}$; the equality above determines the rate of the linear program itself. A comparison theorem of Cohn and Zhao [7] then implies that the ordinary spherical linear program for $K(n)$ cannot attain an exponent smaller than $\frac{1}{2}\log_{2}(2e/\pi)$.

For our bounds, we use the explicit auxiliary functions $f$ constructed in [15] to establish the asymptotic density upper bound, rather than the density-rate theorem alone. The density bound depends on $f(0)/\widehat{f}(0)$ together with a volume factor, whereas the kissing and contact bounds depend on $f(0)/[-f(R)]$, where $R$ is the center-to-center distance at contact. We evaluate the latter ratio using the saddle-point estimate at the target parameter $u_{0}$ in [15], Chapter 1, §4. Lemma 3 below collects the required properties of these functions and the resulting contact-value estimate, with precise references to the global sign and saddle-point estimates on which our argument relies.

All logarithms without a subscript are natural.

The Fourier contact bound

The following bound for average contact degree appears in [1], Theorem 5.3. Its finite-packing assertion also follows from the positive-type bound of Dostert, Kolpakov, and de Oliveira Filho [9], Theorem 4.1, specialized to congruent balls. As noted in [1], it is the Euclidean analogue of Fortier Bourque and Petri’s trace-formula bound [10], Proposition 4.1.

We use the normalization $\widehat{f}(\xi)=\int_{\mathbb{R}^{n}}f(x)e^{-2\pi i\langle x,\xi\rangle}\,\mathrm{d}x$. For radial $f$, write $f(r)$ for its value at a vector of norm $r$.

Lemma 2 ([1], Theorem 5.3; [9], Theorem 4.1). Let $f$ be a real radial Schwartz function on $\mathbb{R}^{n}$, and suppose that, for some $R>0$,

Then every rank-$n$ lattice satisfies $\tau(L)\le f(0)/[-f(R)]$. The same ratio bounds the average contact degree of every nonempty finite packing of balls of radius $R/2$, with contacts at center distance $R$.

Proof. We scale $L$ so that its shortest nonzero vectors have length $R$, equivalently $\min L=R^{2}$. Writing $L^{*}$ for the dual lattice, Poisson summation gives

since $f(R)<0$, this proves the lattice bound.

For a nonempty finite set $C$ of minimum separation at least $R$, Fourier inversion gives

If $M$ counts the ordered pairs at distance $R$, then the last sum is at most $|C|f(0)+Mf(R)$. Thus $M/|C|\le f(0)/[-f(R)]$. $\square$

The auxiliary functions

Fix sufficiently small $\varepsilon>0$, and put

Let $w_{\varepsilon}$ be the compactly supported signed density in [15], Chapter 1, §4,⁴ and define the real function

Thus $g_{\varepsilon}(t)=h_{\varepsilon}(it)$ in the notation of [15], Chapter 1. We collect the analytic input, including the contact value extracted from the [15], Chapter 1 saddle estimate, in the following lemma.

Lemma 3. For every fixed sufficiently small $\varepsilon>0$, there is $n_{0}(\varepsilon)$ such that, for every $n\ge n_{0}(\varepsilon)$, there exist a real radial Schwartz function $f=f_{n,\varepsilon}$ on $\mathbb{R}^{n}$ and a radius $R=R_{n,\varepsilon}>0$ satisfying the hypotheses of Lemma 2, with

In particular, $A_{n,\varepsilon}>0$. The constants implicit in $\asymp_{\varepsilon}$ may depend on $\varepsilon$ but not on $n$.⁵ More precisely, put

Then $V_{n,\varepsilon}\asymp_{\varepsilon}1$ and

Proof. We take $f$ to be the function denoted $f_{-}$ in [15], Chapter 1, §4. The origin value (2) and radius (1) are given in [15], Chapter 1, (42) and (44), respectively. The derivative estimate (4) is [15], Chapter 1, Lemma 4.2, and the global sign conditions are established in [15], Chapter 1, Theorem 4.1. For the contact value, the target parameter is $u=u_{0}$ in that preprint’s notation, and this endpoint is included in [15], Chapter 1, Lemma 4.8. The saddle coefficient $p_{\varepsilon}$ is given in [15], Chapter 1, (38)–(39). In [15], Chapter 1, (70)–(71), the inverse Mellin factor $\lambda/(2\pi)$ and Gaussian factor combine as

giving (3) and (5). The curvature bounds in [15], Chapter 1, Lemma 4.5, (56) give $V_{n,\varepsilon}\asymp_{\varepsilon}1$. Since $p_{\varepsilon}<0$, the contact value is negative and $A_{n,\varepsilon}\asymp_{\varepsilon}\sqrt{n}$. □

We also need the estimate

To verify this, let $w_{\varepsilon,+}$ and $w_{\varepsilon,-}$ be the positive and negative parts of $w_{\varepsilon}$, so that $w_{\varepsilon}=w_{\varepsilon,+}-w_{\varepsilon,-}$. The density construction and [15], Chapter 1, Lemma 4.2 give

for an absolute $c>0$. For $1\le t\le u\le5/4$, we use the inequalities $\sinh(ta)\le ta\cosh(ta)$ and $\cosh(ta)\le e^{(t-1)a}\cosh a$ to obtain

Integration over $[1,u]$, of length $\varepsilon/4$, proves (6).

The kissing number bound

We now estimate the ratio $f(0)/[-f(R)]$ and apply Lemma 2 to obtain the lattice kissing number bound.

Proof of Theorem 1. Fix sufficiently small $\varepsilon>0$. Taking logarithms in (2) and (3) and substituting (1) gives

(The powers of $\pi$ cancel because $2m=\lambda(1+u)$.) Stirling’s formula yields

Since $\log A_{n,\varepsilon}=\frac{1}{2}\log n+O_{\varepsilon}(1)$ and $m/n=(1+u)/4$, we have

Dividing (7) by $n=2\lambda$ therefore gives

Since $u=1+O(\varepsilon)$, equations (4) and (6) then give

Applying Lemma 2 and taking the upper limit as $n\to\infty$ with $\varepsilon$ fixed, we obtain

Letting $\varepsilon\downarrow0$ proves the theorem. $\square$

Average contact degree

The same ratio estimate also bounds average contact degree.

Corollary 4. As $n\to\infty$, uniformly over nonempty finite $C\subset\mathbb{R}^{n}$ with minimum separation at least $1$,

Proof. Fix sufficiently small $\varepsilon>0$, and take $f$ and $R$ from Lemma 3. The scaled set $RC$ has minimum separation at least $R$, and its ordered pairs at distance $R$ correspond exactly to the ordered pairs of $C$ at distance $1$. The finite-packing part of Lemma 2 therefore gives

the estimate for this ratio in the proof of Theorem 1 is independent of $C$. Taking $n\to\infty$ with $\varepsilon$ fixed and then letting $\varepsilon\downarrow0$ proves the stated uniform bound. $\square$

Concluding remarks

Uniformity and contact counts

Explicitly, for every $\eta> 0$ there is $n_{0}(\eta)$ such that, for every $n \ge n_{0}(\eta)$, both $K_{\mathrm{lat}}(n)$ and $\overline{\tau}(C)$ for every eligible $C \subset\mathbb{R}^{n}$ are at most $2^{(\frac{1}{2}\log_{2}(2e/\pi)+\eta)n}$. For a packing of $N$ congruent balls, the contact number—the number of unordered tangent pairs—is therefore at most

uniformly in $N$.

Our bounds apply to lattice kissing numbers and to the average contact degree of arbitrary finite packings of congruent balls. They do not establish the same exponential bound for the maximum contact degree in such a packing, and therefore do not give the same bound for the unrestricted kissing number $K(n)$; they also do not address packings with unequal radii. For background on the contact-number problem, see the Bezdek and Khan [5] survey.

Comparison with the spherical linear program

The exponent in Theorem 1 also gives a lower bound on the exponential rate of the optimal ordinary kissing-number linear-programming bound. Let $K_{n}^{\mathrm{LP}}$ denote the infimum of the upper bounds for $K(n)$ supplied by the two-point spherical linear program of Delsarte, Goethals, and Seidel [8]. Cohn and Zhao [7] specialized to the kissing angle $\pi/3$, proved

their comparison applies in the same dimension $n$.⁶ Combining (8) with $\lim_{n\to\infty}\mathrm{LP}_{n}^{1/n}=\sqrt{e/(2\pi)}$ gives

Thus our bounds for lattice kissing numbers and average contact degree attain an exponent below which the ordinary spherical linear program cannot bound the unrestricted kissing number.⁷ The inequality (9) is a lower bound on the optimal spherical linear programming upper bound $K_{n}^{\mathrm{LP}}$, not a lower bound on the kissing number $K(n)$ itself. Earlier asymptotic lower bounds for the optimal spherical linear-programming bound were obtained by Samorodnitsky [16].

The same constant appears within the spherical-code hierarchy of [15], whose certificates belong to this two-point linear program; see [15]. Specifically, write $\sigma=2\Gamma_{r}$ for a certificate’s threshold and $\Phi=\Phi_{r}$ for its code-size exponent. Then [15] gives

where $\nu$ and $\rho_{K}$ are the probability measures defined there and $D_{\mathrm{KL}}$ denotes relative entropy. For parameters feasible at the kissing angle, $\sigma\ge1/2$; nonnegativity of relative entropy therefore gives $\Phi\ge\frac{1}{2}\log_{2}(2e/\pi)$. This identity makes the same lower bound explicit for those certificates, while the Cohn–Zhao comparison applies to the entire two-point spherical linear program.

Appendix A gives a stronger limitation for the spherical-code hierarchy of [15]: Lemma A.1 shows that, even after spherical-cap optimization and taking the infimum over fixed levels, the hierarchy’s asymptotic kissing exponent remains strictly above ours. This statement concerns that certificate family; it does not establish a strict gap in (9) for the full two-point spherical linear program.

The sharpness argument in [15] Chapter 2, Theorem 8.3 sends $\sigma\to1$, establishing the optimized sphere-packing density exponent. This is compatible with Lemma A.1, which fixes the kissing angle at $\pi/3$.

References

Appendix A. Separation from the OpenAI spherical-code hierarchy

In this appendix, we show that the asymptotic kissing-number exponents supplied by the hierarchy of [15] Chapter 2 remain uniformly separated from the exponent in Theorem 1, even after spherical-cap optimization and taking the infimum over fixed levels. Starting from the relative-entropy identity of [15], Chapter 2, Proposition 8.1, we use Jensen’s inequality to bound the mass of a fixed interval and thereby obtain an explicit positive lower bound on the relative entropy.

Lemma A.1. Put

The infimum of the fixed-level asymptotic kissing-number exponents supplied by the hierarchy of [15], Chapter 2, Theorem 1.2, over all finite levels and the spherical-cap parameters in [15], Chapter 2, (89), is at least

Here the level is fixed before the dimension tends to infinity, and the infimum is then taken over the resulting asymptotic rates. The argument does not address levels that vary with the dimension.

Proof of Lemma A.1. For a finite-level certificate, let $x_i\geq0$ and $R_i>0$ be the quadratic coordinates and residues in [15], Chapter 2, Proposition 8.1, (102), so that $\sum_i R_i=1$. Define

where $\delta_w$ denotes unit point mass at $w$, and put

In particular, $k_0(v)=1$ and $k_1(v)=0$. Under $t=(1-v^2)/4$, the reference measures in [15], Chapter 2, Proposition 8.1 become

The kernel satisfies

We define the normalizing constant

the kernel identity gives

By [15], Chapter 2, (102)–(104), the certificate’s threshold $\sigma$ and exponent $\Phi$ satisfy

We now enlarge the class under consideration by allowing arbitrary probability measures $\mu$ on $[0,1]$, including endpoint atoms, rather than only those arising from finite-level certificates. For these measures, define $F$ and $Z$ as above and $\sigma$ and $\Phi$ by the same formulas in (10). The kernel identity ensures that $Z=1-\sigma$ continues to hold. We set $\Phi=+\infty$ if $F\equiv0$.

We first include cap optimization and reduce to $\sigma=1/2$. A certificate used at an inner-product threshold $0\leq s\leq1/2$ is feasible when $\sigma\geq s$; cap reduction adds $\frac{1}{2}\log_2(2(1-s))$ to its exponent.

Set $q=1/(2(1-s))$ and replace $\mu$ by $q\mu+(1-q)\delta_{1}$. This replaces $F$ by $qF$, adds exactly that cap cost, and replaces the normalizing constant by

hence, the new measure has mean at least $1/2$. Direct use at the kissing angle is the case $s=1/2$, $q=1$; the endpoint $s=0$ is also covered by taking $\mu=\delta_{0}$.

We may further restrict to measures of mean $1/2$. Indeed, for any probability measure $\mu$ with mean $\sigma\ge1/2$, put

Then $\widetilde{\mu}$ has mean $1/2$. Since $k_{0}(v)=1$ and $0\le F(v)\le1$, its associated function $\widetilde{F}$ satisfies

Thus its associated exponent satisfies $\widetilde{\Phi}\le\Phi$. A lower bound for the exponent of every probability measure of mean $1/2$ therefore also bounds the exponent of every probability measure of mean at least $1/2$, including the measures obtained by the cap reduction above.

For such a measure, $P=2F\rho$ is a probability measure, and the relative-entropy identity becomes

Indeed, $\int_{0}^{1}\log(\mathrm{d}\nu/\mathrm{d}\rho)\,\mathrm{d}\nu=\log\pi-1$, which gives (11) directly from (10). The logarithms are integrable: $k_{w}(v)\ge(1-w)(1-v^{2})$ gives $F(v)\ge(1-v^{2})/2$.

For $0\le v\le1/2$, the function $w\mapsto k_{w}(v)$ is concave, since

Jensen’s inequality gives

For $E=[0,1/2]$, we have $\nu(E)=1/2$ and consequently

The integral follows by setting $v=\sin\theta$; the strict inequality follows from the fact that $\tan(\pi/12)=2-\sqrt{3}<1/(2\sqrt{3})$. Applying the log-sum inequality to the partition $\{E,E^{c}\}$ then gives

Substitution in (11) proves the claimed bound. (It is independent of the level and cap parameter, so taking their infimum preserves it.) $\square$