This is the ninth thread for the Polymath8b project to obtain new bounds for the quantity $\displaystyle H_m := \liminf_{n \rightarrow\infty} (p_{n+m} - p_n),$

either for small values of ${m}$ (in particular ${m=1,2}$) or asymptotically as ${m \rightarrow \infty}$. The previous thread may be found here. The currently best known bounds on ${H_m}$ can be found at the wiki page.

The focus is now on bounding ${H_1}$ unconditionally (in particular, without resorting to the Elliott-Halberstam conjecture or its generalisations). We can bound ${H_1 \leq H(k)}$ whenever one can find a symmetric square-integrable function ${F}$ supported on the simplex ${{\cal R}_k := \{ (t_1,\dots,t_k) \in [0,+\infty)^k: t_1+\dots+t_k \leq 1 \}}$ such that $\displaystyle k \int_{{\cal R}_{k-1}} (\int_0^\infty F(t_1,\dots,t_k)\ dt_k)^2\ dt_1 \dots dt_{k-1} \ \ \ \ \ (1)$ $\displaystyle > 4 \int_{{\cal R}_{k}} F(t_1,\dots,t_k)^2\ dt_1 \dots dt_{k-1} dt_k.$

Our strategy for establishing this has been to restrict ${F}$ to be a linear combination of symmetrised monomials ${[t_1^{a_1} \dots t_k^{a_k}]_{sym}}$ (restricted of course to ${{\cal R}_k}$), where the degree ${a_1+\dots+a_k}$ is small; actually, it seems convenient to work with the slightly different basis ${(1-t_1-\dots-t_k)^i [t_1^{a_1} \dots t_k^{a_k}]_{sym}}$ where the ${a_i}$ are restricted to be even. The criterion (1) then becomes a large quadratic program with explicit but complicated rational coefficients. This approach has lowered ${k}$ down to ${54}$, which led to the bound ${H_1 \leq 270}$.

Actually, we know that the more general criterion $\displaystyle k \int_{(1-\epsilon) \cdot {\cal R}_{k-1}} (\int_0^\infty F(t_1,\dots,t_k)\ dt_k)^2\ dt_1 \dots dt_{k-1} \ \ \ \ \ (2)$ $\displaystyle > 4 \int F(t_1,\dots,t_k)^2\ dt_1 \dots dt_{k-1} dt_k$

will suffice, whenever ${0 \leq \epsilon < 1}$ and ${F}$ is supported now on ${2 \cdot {\cal R}_k}$ and obeys the vanishing marginal condition ${\int_0^\infty F(t_1,\dots,t_k)\ dt_k = 0}$ whenever ${t_1+\dots+t_k > 1+\epsilon}$. The latter is in particular obeyed when ${F}$ is supported on ${(1+\epsilon) \cdot {\cal R}_k}$. A modification of the preceding strategy has lowered ${k}$ slightly to ${53}$, giving the bound ${H_1 \leq 264}$ which is currently our best record.

However, the quadratic programs here have become extremely large and slow to run, and more efficient algorithms (or possibly more computer power) may be required to advance further.