Threshold complexes and connections to number theory (Q2862227)

This very interesting paper studies threshold complexes or, equivalently, quota complexes, which arise naturally in many areas, for example in voting theory in political science. Surprisingly the quota complexes allow topological interpretation of many important problems of number theory such as the Twin Prime Conjecture, Goldbach Conjecture, Lehmer Conjecture, the Riemann Hypothesis and others.NEWLINENEWLINEGiven a vertex set \(V\), a weight function \(w: V\to \mathbb R\), and a quota \(q\in \mathbb R\), the \textit{(scalar) quota complex} \(X[w:q]\) is the simplicial complex on the vertex set \(V\) such that a face \(F=[v_0, \dots, v_n]\) is in \(X[w:q]\) iff \(\sum_{i=0}^n w(v_i) <q\).NEWLINENEWLINEIf the weight function and the quota \(q\) are vector valued, \(\hat w: V\to {\mathbb R}^s\) and \(\hat q\in {\mathbb R}^s\), then \textit{the vector-valued quota complex} \(X[\hat w:\hat q]\) is defined as the simplicial complex with the vertex set \(V\) such that a face \(F=[v_0, \dots, v_n]\) is in \(X[\hat w:\hat q]\) iff \(\sum_{i=0}^n w_j(v_i) <q_j\) for some \(j=1, \dots, s\). In other words, NEWLINE\[NEWLINEX[\hat w:\hat q]= \cup_{j=1}^s X[w_j:q_j].NEWLINE\]NEWLINENEWLINENEWLINEThe authors show that any simplicial complex is isomorphic to a vector-valued quota complex. On the other hand, the scalar-valued quota complexes are very specific, for example they have homotopy type of a wedge of spheres, and the authors give an explicit description of these spheres.NEWLINENEWLINEIf \(V\) is a set of voters, then a subset \(C\subset V\) is called a losing coalition if the voters \(C\) are not able to force an initiative to pass. A subset of a losing coalition is also a losing coalition and hence we obtain a simplicial complex encoding the properties of the voting system. In voting theory it is shown that every voting system can be weighted so that the simplicial complex of losing coalitions is a quota complex.NEWLINENEWLINELet \(V\) be the set of primes and let \(w: V\to {\mathbb R}\) be the function \(w(p)=p\), i.e. the weight of a prime equals its value. This defines a specific quota complex denoted \(Prime(q)\). The authors show that the Twin Prime Conjecture is equivalent to the statement that \(Prime(q)\) and \(Prime(q+2)\) are both disconnected complexes for infinitely many values of \(q\). The Goldbach Conjecture is equivalent to the statement that \(Prime(q)\) has a non-simply connected component for all \(q\geq 6\) with \(q\) not equal to twice an odd prime.NEWLINENEWLINEThe \(LogPrime(q)\) is a quota complex on the set of primes with the weight function \(w(p)=\ln(p)\). The Riemann Hypothesis is equivalent to the statement that for any \(\epsilon >0\) there is a constant \(C>0\) such that NEWLINE\[NEWLINE\ln\left(\left| \chi(LogPrime(q))\right|\right) \leq (0.5+\epsilon)q+\ln(C)NEWLINE\]NEWLINE for all \(q\) large enough. This statement is related to a result of \textit{A. Björner} [Adv. Appl. Math. 46, No. 1--4, 71--85 (2011; Zbl 1230.05300)].NEWLINENEWLINEThe authors also study random quota complexes when the weights are random variables with certain properties.