Orbital maximin is bounded below by constant fraction of number of ordered pairs of distinct elements for solvable groups
Let be a set of size , with . Consider the orbital maximin problem for the action of solvable groups on , i.e., we want to find the maximum possible value of the smallest orbital size under a solvable group action on . The total number of ordered pairs of distinct elements is . The claim is that the following equivalent statements are true:
- There is a constant such that the orbital maximin is for all .
- There is a constant such that, for all sufficiently large , the orbital maximin is .
We prove (2).
The general construction: additive partitioning
Consider an expression for as a sum of primes:
Consider now the group:
where is the general affine group (also called the affine general linear group) over the prime field with elements. In other words, it is the semidirect product of the additive group of the field and the multiplicative group of the field. Note that each is a metabelian group, hence so is .
We consider now the following action of on a set of size . First, divide into subsets of size each. Identify each with the underlying set of the field of elements. Now, the action of an element of on the piece is the action of its coordinate viewed as an element of the affine group acting on the field.
The orbits of elements are the sets , and there are two kinds of orbitals:
- The orbitals where both elements are in the same : These have size because the action of the general affine group is a doubly transitive group action.
- The orbitals where the two elements are in different s, say and : These have size .
The minimum of all these turns out to be where is the minimum of the s. Therefore, if we can guarantee a partition into primes where each of the primes is for some constant , we will be done.
Result from additive number theory
The result now follows from Haselgrove's strengthening of Vinogradov's theorem, which states that any odd integer can be divided into three primes that are , and an easy corollary which shows that any even integer can be divided into three primes that are . This would show that any constant $c_0 > 16</math> will work. Also, we see that our choice of has polycyclic breadth at most eight.
If strong versions of Goldbach's conjecture are true, we can push it down to any constant .