Some equivalents of the Erdős sum of reciprocals conjecture (Q1106266)

A number of equivalent statements to the ``Erdős sum of reciprocals'' conjecture (each subset A of positive integers with \(\sum_{x\in A}1/x=\infty\) contains arbitrarily long arithmetic progressions) is given. One part of these statements is of combinatorial kind and the other part uses the algebraic structure of the Stone-Čech compactification \(\beta\) \({\mathbb{N}}\) of the positive integers \({\mathbb{N}}.\) For \(k\in {\mathbb{N}}\), \(k\geq 3\) set \({\mathcal S}_ k=\{A\subseteq {\mathbb{N}}:\) there do not exist \(a,d\in {\mathbb{N}}\) with \(\{a+td:\) \(t\in \{0,1,...,k- 1\}\} \subseteq A\}\). Let SR(k) be the statement ``whenever \(A\subseteq {\mathbb{N}}\) and \(\sum_{x\in A}1/x=\infty\), there exist \(a,d\in {\mathbb{N}}\) with \(\{a+td:\) \(t\in \{0,1,...,k-1\}\}\subseteq A''\). Let B(k) be the statement ``there exists \(z\in {\mathbb{R}}\) (the set of real numbers) such that \(\sum_{x\in A}1/x<z\) whenever \(A\in {\mathcal S}_ k''\). Further similar statements \(M_ 1(k)\), \(M_ 2(k)\), \(N_ 1(k)\), \(N_ 2(k)\) and P(k) are defined. The results of the first part are contained in the following theorem (Theorem 2.6): ``Let \(k\in {\mathbb{N}}\) with \(k\geq 3\). The statements SR(k), \(M_ 1(k)\), \(M_ 2(k)\), B(k), \(N_ 1(k)\), \(N_ 2(k)\), and P(k) are pairwise equivalent''. (The author remarks that the equivalence of SR(k) and B(k) is due to Erdős.) The following notions concerning the Stone-Čech compactification \(\beta\) \({\mathbb{N}}\) are used (p denotes an ultrafilter on \({\mathbb{N}}):\) (a) \({\mathcal A}{\mathcal P}=\{p\in \beta {\mathbb{N}}:\) for all \(A\in p\) and all \(k\in {\mathbb{N}}\) there exist \(a,d\in {\mathbb{N}}\) with \(\{a+td:\) \(t\in \{0,1,...,k- 1\}\} \subseteq A\}\), (b) \({\mathcal D}=\{p\in \beta {\mathbb{N}}:\) for all \(A\in p\), \(\sum_{x\in A}1/x=\infty \}\), (c) \({\mathcal M}=\{p\in \beta {\mathbb{N}}:\) for all \(A\in p\), there exist increasing sequences \(<t_ n>^{\infty}_{n=1}\) and \(<s>^{\infty}_{n=1}\) in \({\mathbb{N}}\) such that \(\sum^{\infty}_{n=1}1/s_ n=\infty\) and \(\{t_ n+s_ m:\) \(n,m\in {\mathbb{N}}\) and \(m\leq n\} \subseteq A\}\), (d) \({\mathcal S}=\{p\in \beta {\mathbb{N}}:\) for all \(A\in p\), there exist increasing sequences \(<t_ n>^{\infty}_{n=1}\) and \(<s_ n>^{\infty}_{n=1}\), and a non- decreasing sequence \(<a_ n>^{\infty}_{n=1}\) in \({\mathbb{N}}\) such that \(\sum^{\infty}_{n=1}1/s_ n=\infty\) and \(\{t_ n+a_ n\cdot s_ m:\) \(n,m\in {\mathbb{N}}\) and \(m\leq n\} \subseteq A\}.\) The second part of the equivalent statements is contained in Theorem 4.3: ``The following statements are equivalent: for all \(k\in {\mathbb{N}}\), SR(k), \({\mathcal D}\subseteq {\mathcal A}{\mathcal P}\), \({\mathcal M}\subseteq {\mathcal A}{\mathcal P}\), \({\mathcal S}={\mathcal A}{\mathcal P}''\).