On dilution and Cesàro summation

The problem whether a real sequence $({s_i})$ has a dilution which is $(C,1)$ summable to a number $s$ is transformed by means of two sequences measuring the oscillation of $({s_i})$ about $s$. (If it does not oscillate, the condition, known, is that $s$ is a limit point of $({s_i})$.) For the $j$th consecutive block of ${s_i}$ on one side of $s,{\alpha _j}$ is the minimum of their distances from $s,{\beta _j}$ the sum of distances. Then there must exist positive numbers ${p_j}$ such that ${\beta _j} + {p_j}{\alpha _j} = o({p_1} + \cdots + {p_{j - 1}})$. The necessary condition and the sufficient condition coincide for very smooth sequences at ${\alpha _i}\log {\beta _i} = o(i)$.