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1. Miscellaneous problems


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      Problem 1.05.

      [Folklore] Prove the universal optimality of $A_2$.


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          Conjecture 1.1.

          [Doug Hardin] The second-order term of the minimal Riesz energy on $N$ points on the sphere: \[ \sum_{i=1}^N \|x_i-x_j\|^{-s} \longrightarrow \min, \qquad s.t.\ x_1,\ldots x_N \subset \mathbb S^d \] is of the order $N^{1+s/d}$ for $0< s < d$. That is, if $x_1^*,\ldots,x_N^*$ attain the minimum, then \[ \sum_{i=1}^N \|x_i^*-x_j^*\|^{-s} - N^2\iint_{\mathbb S^d} \|x-y\|^{-s}\,d\sigma(x)d\sigma(y) = C_{s,d} N^{1+s/d} + o(N^{1+s/d}). \]

            •     Consider the asymptotic expansion of the minimal logarithmic energy of $N$ charges on the 2-sphere: $$ \mathcal E_n = C_1N^2 + C_2 N \log N +C_3N +\boldsymbol{?} + C_4\log N $$ Values of $C_1$, $C_2$, $C_3$ are known, cf. Bétermin-Sandier [MR3742809].

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              Problem 1.15.

              [Ed Saff] What is the order of the next unknown term in the asymptotic expansion of minimal logarithmic energy? Candidates: $ \sqrt N\log N$, $\log N$.

              What is the information this next term encodes? E.g., is it related to the behavior of scars from the previous problem?


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                  Problem 1.2.

                  [Carlos Beltrán] Find good upper/lower bound for the minimum of the quantity $$ E(z_1,\ldots, z_N):= \frac{\|(z_1,\ldots, z_N)\|_2 }{\min_{i\neq j}|z_i - z_j|} $$ over $N$ complex numbers $z_i$. Here $\|\cdot\|_2$ denotes the Hermitian norm of a vector in $\mathbb C^N$.
                    1. Remark. Solved in arXiv:2105.07922


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                          Problem 1.25.

                          [Sasha Reznikov] Prove poppy-seed bagel theorem for fractals. That is, determine for which d-dimensional fractals in $\mathbb R^p$ the asymptotics of the minimal Riesz energy with $s>d$ exist. Also, when do the minimizers converge to the uniform distribution on the fractal?


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                              Problem 1.3.

                              [Mircea Petrache] Study universality of lattices on smaller classes of functions (instead of completely monotonic functions). This would also be relevant to the study of self-assembly phenomena in physics.

                              Some previous results: Bétermin-Petrache [ MR4038122], Cohn-Kumar Algorithmic design of self-assembling structures.

                                •     A preprint of Dragnev-Musin shows that minimizers of logarithmic energy among the d+3 point configurations on d-dimensional sphere are pairs of orthogonal simplexes.

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                                  Problem 1.35.

                                  [Peter Dragnev] Characterize the minimizers consisting of d+3 points on the d-sphere, if the interaction is given by the Riesz kernel.

                                  More specifically, extend the results of Schwartz to larger values of $s$. Bonus points if his results can be recovered without computer assistance.


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                                      Problem 1.4.

                                      [Sergiy Borodachov] Find the asymptotics for the maximal Riesz polarization / minimal Riesz energy on a d-rectifiable compact set A in $\mathbb R^p$, assuming $s=d$.

                                        •     Reznikov-Volberg-Saff [MR3774346] establish that an N-point configuration that has optimal polarization on a set with boundary of measure zero will contain no more than $k(d)$ points in any ball of radius $cN^{-1/d}$. This can be referred to as weak separation, in contrast with the strong separation results, which give: no two points can be in a ball of radius $cN^{-1/d}$.

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                                          Problem 1.45.

                                          [Sergiy Borodachov] Prove weak separation for the maximal polarization configurations on more general sets.


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                                              Problem 1.5.

                                              [Sergiy Borodachov] Can linear programming be applied to max-min polarization?

                                                  Cite this as: AimPL: Minimal energy problems with Riesz potentials, available at http://aimpl.org/energyrieszV.