nt.number theory - Erdos Conjecture on arithmetic progressions

There is a way you can recast Erdős conjecture into a statement about certain inclusions among various compact left and two-sided ideals. Such topological-algebraic statements, and a few combinatorial statements, are proved by Neil Hindman in

"Some Equivalents of the Erdős Sum of Reciprocals Conjecture." European Journal of Combinatorics (1988) 9, no. 1, 39 -- 47.

Here is a brief sample of one of these topological-algebraic statements. Let $betamathbb{N}$ denote the Stone-Čech compactification of the discrete space $mathbb{N}$. We can extend the usual addition and multiplication operations on $mathbb{N}$ to $betamathbb{N}$ to make $(betamathbb{N}, +)$ and $(betamathbb{N}, cdot)$ both into compact right-topological semigroups. (Right topological semigroup means that $(betamathbb{N}, +)$ and $(betamathbb{N}, cdot)$ are both semigroups and for all $p$, $q in betamathbb{N}$ the maps $p mapsto p+q$ and $p mapsto pcdot q$ are continuous.) To see how to actually perform this extension you can read section 3, pgs. 23-28, of this pdf document by Vitaly Bergelson. (However, Bergelson's construction makes $(betamathbb{N}, +)$ into a compact left-topological semigroup.)

Let $L subseteq betamathbb{N}$. We say $L$ is a left ideal of $(betamathbb{N}, +)$ if $L$ is nonempty and $betamathbb{N} + L subseteq L$. We define a right ideal of $(betamathbb{N}, +)$ dually, and a (two-sided) ideal is both a left and right ideal. We define left, right, and two-sided ideals of $(betamathbb{N}, cdot)$ by simply replacing "$+$" with "$cdot$" above.

Now define the following two subsets of $betamathbb{N}$:

• $mathcal{AP} = {p in betamathbb{N} : A hbox{ contains APs of arbitrary length for all } A in p }$


• $mathcal{D} = {p in betamathbb{N} : sum_{nin A} 1/n = infty hbox{ for all } A in p}$

It's known that $mathcal{AP}$ is a compact two-sided ideal of $(betamathbb{N}, +)$ and $(betamathbb{N}, cdot)$, and that $mathcal{D}$ is a compact left ideal of $(betamathbb{N}, +)$ and $(betamathbb{N}, cdot)$. Therefore (part of) the main result of Hindman's paper is the

Theorem. The following statements are equivalent.

(a) If $Asubseteq mathbb{N}$ and $sum_{n in A} 1/n = infty$, then A contains APs of arbitrary length.

(b) $mathcal{D} subseteq mathcal{AP}$.

Of course the point here is that since $mathcal{D}$ is a left ideal and $mathcal{AP}$ is a two-sided ideal you would hope to have some nice theorems about inclusion relationships among various compact left, right, and two-sided ideals in $betamathbb{N}$ to lean on. As far as I know, no one has attempted to attack Erdős conjecture from this topological-algebraic viewpoint.

Just to further illustrate the difficulties involved, let me mention that currently there is not even a "purely" topological-algebraic proof of Szemerédi's Theorem yet!

Let $Delta = {p in betamathbb{N} : overline{d}(A) > 0 hbox{ for all } A in p}$ and let $Delta^* = {p in betamathbb{N} : d^*(A) > 0 hbox{ for all } A in p}$. Here $overline{d}$ and $d^*$ are the upper asymptotic density and Banach Density. It's known that $Delta$ is a compact left ideal of $(betamathbb{N},+)$, and $Delta^*$ is a compact two-sided ideal of $(betamathbb{N}, +)$ and a compact left ideal of $(betamathbb{N}, cdot)$. In the above paper, Hindman shows that Szemerédi's Theorem is equivalent to each of the inclusions $Delta subseteq mathcal{AP}$ and $Delta^* subseteq mathcal{AP}$.

However, one possible approach to show Szemerédi's Theorem in a topological-algebraic "way" was shown not to work in the paper "Subprincipal Closed Ideals in $betamathbb{N}$" by Dennis Davenport and Hindman. In this paper, they show that $Delta^*$ intersects every closed ideal of $(betamathbb{N},+)$; but, beyond that, not enough is known about inclusions among compact ideals to prove, algebraically, that $Delta^* subseteq mathcal{AP}$.