Poppy-seed bagel theorem

For a parameter ${\displaystyle s>0}$ and an ${\displaystyle N}$-point set ${\displaystyle \omega _{N}=\{x_{1},\ldots ,x_{N}\}\subset \mathbb {R} ^{p}}$, the ${\displaystyle s}$-energy of ${\displaystyle \omega _{N}}$ is defined as follows:

$${\displaystyle E_{s}(\omega _{N}):=\sum _{i=1,\ldots ,N}\sum _{\stackrel {j=1,\ldots ,N}{j\not =i}}{\frac {1}{|x_{i}-x_{j}|^{s}}}}$$

For a compact set ${\displaystyle A}$ we define its minimal ${\displaystyle N}$-point ${\displaystyle s}$-energy as

$${\displaystyle {\mathcal {E}}_{s}(A,N):=\min E_{s}(\omega _{N}),}$$

where the minimum is taken over all ${\displaystyle N}$-point subsets of ${\displaystyle A}$; i.e., ${\displaystyle \omega _{N}\subset A}$. Configurations ${\displaystyle \omega _{N}}$ that attain this infimum are called ${\displaystyle N}$-point ${\displaystyle s}$-equilibrium configurations.

We consider compact sets ${\displaystyle A\subset \mathbb {R} ^{p}}$ with the Lebesgue measure ${\displaystyle \lambda (A)>0}$ and ${\displaystyle s\geqslant p}$. For every ${\displaystyle N\geqslant 2}$ fix an ${\displaystyle N}$-point ${\displaystyle s}$-equilibrium configuration ${\displaystyle \omega _{N}^{*}=\{x_{1,N},\ldots ,x_{N,N}\}}$. Set

$${\displaystyle \mu _{N}:={\frac {1}{N}}\sum _{i=1,\ldots ,N}\delta _{x_{i,N}},}$$

where ${\displaystyle \delta _{x}}$ is a unit point mass at point ${\displaystyle x}$. Under these assumptions, in the sense of weak convergence of measures,

$${\displaystyle \mu _{N}{\stackrel {*}{\rightarrow }}\mu ,}$$

where ${\displaystyle \mu }$ is the Lebesgue measure restricted to ${\displaystyle A}$; i.e., ${\displaystyle \mu (B)=\lambda (A\cap B)/\lambda (A)}$. Furthermore, it is true that

$${\displaystyle \lim _{N\to \infty }{\frac {{\mathcal {E}}_{s}(A,N)}{N^{1+s/p}}}={\frac {C_{s,p}}{\lambda (A)^{s/p}}},}$$

where the constant ${\displaystyle C_{s,p}}$ does not depend on the set ${\displaystyle A}$ and, therefore,

$${\displaystyle C_{s,p}=\lim _{N\to \infty }{\frac {{\mathcal {E}}_{s}([0,1]^{p},N)}{N^{1+s/p}}},}$$

where ${\displaystyle [0,1]^{p}}$ is the unit cube in ${\displaystyle \mathbb {R} ^{p}}$.

Near minimal ${\displaystyle s}$-energy 1000-point configurations on a torus (${\displaystyle d=2}$)

${\displaystyle s=0.01<d}$

${\displaystyle s=1<d}$

${\displaystyle s=2=d}$

Consider a smooth ${\displaystyle d}$-dimensional manifold ${\displaystyle A}$ embedded in ${\displaystyle \mathbb {R} ^{p}}$ and denote its surface measure by ${\displaystyle \sigma }$. We assume ${\displaystyle \sigma (A)>0}$. Assume ${\displaystyle s\geqslant d}$ As before, for every ${\displaystyle N\geqslant 2}$ fix an ${\displaystyle N}$-point ${\displaystyle s}$-equilibrium configuration ${\displaystyle \omega _{N}^{*}=\{x_{1,N},\ldots ,x_{N,N}\}}$ and set

$${\displaystyle \mu _{N}:={\frac {1}{N}}\sum _{i=1,\ldots ,N}\delta _{x_{i,N}}.}$$

Then,[2][3] in the sense of weak convergence of measures,

$${\displaystyle \mu _{N}{\stackrel {*}{\rightarrow }}\mu ,}$$

where ${\displaystyle \mu (B)=\sigma (A\cap B)/\sigma (A)}$. If ${\displaystyle H^{d}}$ is the ${\displaystyle d}$-dimensional Hausdorff measure, then[2][4]

$${\displaystyle \lim _{N\to \infty }{\frac {{\mathcal {E}}_{s}(A,N)}{N^{1+s/d}}}=2^{s}\alpha _{d}^{-s/d}\cdot {\frac {C_{s,d}}{(H^{d}(A))^{s/d}}},}$$

where ${\displaystyle \alpha _{d}=\pi ^{d/2}/\Gamma (1+d/2)}$ is the volume of a d-ball.

For ${\displaystyle p=1}$, it is known[4] that ${\displaystyle C_{s,1}=2\zeta (s)}$, where ${\displaystyle \zeta (s)}$ is the Riemann zeta function. The following connection between the constant ${\displaystyle C_{s,p}}$ and the problem of Sphere packing is known:[5]

$${\displaystyle \lim _{s\to \infty }(C_{s,p})^{1/s}={\frac {1}{s}}\left({\frac {\alpha _{p}}{\Delta _{p}}}\right)^{1/p},}$$

where ${\displaystyle \alpha _{p}}$ is the volume of a p-ball and

$${\displaystyle \Delta _{p}=\sup \rho ({\mathcal {P}}),}$$

where the supremum is taken over all families ${\displaystyle {\mathcal {P}}}$ of non-overlapping unit balls such that the limit

$${\displaystyle \rho ({\mathcal {P}})=\lim _{r\to \infty }{\frac {\lambda \left([-r,r]^{p}\cap \bigcup _{B\in {\mathcal {P}}}B\right)}{(2r)^{p}}}}$$

exists.

• Hausdorff dimension
• Geometric measure theory
• Sphere packing
• Riemann zeta function