Progressions from Divergent Reciprocals, at Every $k$

The Erdos–Tur\\'an conjecture assumes divergence of a reciprocal sum, whereas Szemer\\'edi's theorem assumes positive density. The former is weaker: the primes have divergent reciprocal sum and density zero. For every $k$, we reduce the conjecture to a bound for the maximal size $r_k(N)$ of a $k$-term-progression-free subset of $[0,N)$. If $r_k(N) CN/( N)^1+$ for some $>0$, then every set with divergent reciprocal sum contains a $k$-term progression. The only step requiring attention for general $k$ is translation invariance, which is proved directly. We also identify the threshold supplied by this argument. An exponent of one leads to the harmonic series, while an exponent greater than one yields a convergent comparison series. Similar threshold tests are given for log-log and stretched-log bounds. Behrend's construction shows that no power saving is available at $k=3$.