{
  "H0": null,
  "H1": "There exist an absolute constant \\(C>0\\) and an integer \\(\\ell_0\\ge2\\) such that, for every integer \\(\\ell\\ge\\ell_0\\), there exists a totally real cyclic cubic field \\(F/\\mathbb Q\\) with the following properties.\n\nLet \\(F^{\\mathrm{ur},3}\\) be the maximal everywhere-unramified pro-\\(3\\) extension of \\(F\\), and put \\(G=\\operatorname{Gal}(F^{\\mathrm{ur},3}/F)\\). Then\n\\[\nd(G)\\ge\\ell-1,\n\\qquad\n\\log\\operatorname{rd}(F)\\le C\\ell\\log\\ell.\n\\]\n\nHere \\(d(G)\\) is the minimum number of topological generators of \\(G\\).\n\nHere \\(\\operatorname{rd}(L)=|\\operatorname{Disc}(L)|^{1/[L:\\mathbb Q]}\\) is the root discriminant of a number field \\(L\\), and all logarithms are natural.",
  "H2": "There exist an absolute constant \\(C>0\\) and an integer \\(\\ell_0\\ge2\\) such that, for every integer \\(\\ell\\ge\\ell_0\\), putting\n\\[\nt=\\left\\lfloor\\frac{(\\ell-1)^2}{100}\\right\\rfloor,\n\\]\nthere exist a totally real cyclic cubic field \\(F/\\mathbb Q\\), distinct rational primes \\(q_1,\\ldots,q_t\\), and a tower\n\\[\nF=F_0\\subset F_1\\subset F_2\\subset\\cdots\n\\]\nsatisfying all the following properties.\n\n1. Every \\(q_b\\) is congruent to \\(1\\pmod4\\) and splits completely in every \\(F_j\\), for every \\(j\\ge0\\).\n\n2. Every \\(F_j/F\\) is finite Galois and everywhere unramified, and its Galois group is a \\(3\\)-group. Every \\(F_j\\) is totally real.\n\n3. Writing \\(f_j=[F_j:\\mathbb Q]\\), one has\n\\[\nf_j\\longrightarrow\\infty,\n\\qquad\n\\operatorname{rd}(F_j)=\\operatorname{rd}(F),\n\\qquad\n\\log\\operatorname{rd}(F)\\le C\\ell\\log\\ell.\n\\]\n\nFor each fixed \\(\\ell\\), the primes \\(q_1,\\ldots,q_t\\) are chosen once and are independent of \\(j\\).\n\nHere \\(\\operatorname{rd}(L)=|\\operatorname{Disc}(L)|^{1/[L:\\mathbb Q]}\\) is the root discriminant of a number field \\(L\\), and all logarithms are natural.",
  "H3": "There exists an absolute constant \\(C\\ge1\\) such that, for every totally real number field \\(F\\) of degree \\(f=[F:\\mathbb Q]\\), the field\n\\[\nK=F(\\mathrm i),\\qquad \\mathrm i^2=-1,\n\\]\nsatisfies\n\\[\n\\operatorname{rd}(K)\\le2\\operatorname{rd}(F),\n\\qquad\nh(K)\\le\\bigl(C\\operatorname{rd}(F)\\bigr)^{2f}.\n\\]\n\nHere \\(h(K)\\) is the ideal class number of the ring of integers of \\(K\\). The constant \\(C\\) is independent of \\(F\\) and \\(f\\).\n\nHere \\(\\operatorname{rd}(L)=|\\operatorname{Disc}(L)|^{1/[L:\\mathbb Q]}\\) is the root discriminant of a number field \\(L\\), and all logarithms are natural.",
  "H4": "There exist a fixed integer \\(m\\ge1\\), a fixed real number \\(\\gamma>0\\), a sequence of number fields \\((K_j)_{j\\ge1}\\), and finite sets \\(U_j\\subset K_j\\), such that, writing \\(d_j=[K_j:\\mathbb Q]\\), one has\n\\[\nd_j\\longrightarrow\\infty,\n\\qquad\nU_j\\subset m^{-1}\\mathcal O_{K_j},\n\\qquad\n|U_j|\\ge\\exp(\\gamma d_j).\n\\]\nFor every \\(j\\ge1\\), every \\(u\\in U_j\\), and every field embedding \\(\\sigma:K_j\\hookrightarrow\\mathbb C\\),\n\\[\n|\\sigma(u)|=1.\n\\]\n\nHere \\(\\mathcal O_{K_j}\\) is the ring of integers of \\(K_j\\), and \\(m^{-1}\\mathcal O_{K_j}=\\{a/m:a\\in\\mathcal O_{K_j}\\}\\). The same constants \\(m\\) and \\(\\gamma\\) apply to every \\(j\\).",
  "H5": "For a finite set \\(P\\subset\\mathbb R^2\\), let\n\\[\n\\nu(P)=\\#\\bigl\\{\\{x,y\\}\\subset P:x\\ne y,\\ \\|x-y\\|_2=1\\bigr\\},\n\\]\nwhere \\(\\|\\cdot\\|_2\\) is the Euclidean norm. For every positive integer \\(n\\), define\n\\[\n\\nu(n)=\\max\\{\\nu(P):P\\subset\\mathbb R^2,\\ |P|=n\\}.\n\\]\nThere exist a fixed real number \\(\\delta>0\\) and a sequence of finite planar sets \\((P_j)_{j\\ge1}\\) such that, writing \\(n_j=|P_j|\\),\n\\[\nn_j\\longrightarrow\\infty,\n\\qquad\n\\nu(P_j)\\ge n_j^{1+\\delta}\\quad\\text{for every }j\\ge1.\n\\]\nConsequently, infinitely many distinct positive integers \\(n\\) satisfy\n\\[\n\\nu(n)\\ge n^{1+\\delta}.\n\\]\nThe same \\(\\delta\\) applies to every set in the sequence."
}