The h-measure exceptional-set conjecture for gap power series

Let $\Lambda=(\lambda_n)$ be the frequency sequence and let $L$ and $L^{-}$ denote the classes of auxiliary functions used in the paper. For $\varphi\in L$, define

$$D_{\varphi}(\Lambda)=\big\{F\in D(\Lambda): (\exists n_0)(\forall n\geq n_0)\ [|a_n|\leq \exp\{-\lambda_n\varphi(\lambda_n)\}]\big\}.$$

Let $h$ be the function defining the $h$-measure, and let relation (3) denote the paper's asymptotic relation for the gap power series, uniformly in $y\in\mathbb{R}$. The h-measure exceptional-set conjecture. If

$$\sum_{n=0}^{+\infty}\frac{h^{\prime}(\varphi(\lambda_n))}{\lambda_{n+1}-\lambda_n}<+\infty,$$

then for every $F\in D_{\varphi}(\Lambda)$, relation (3) holds as $x\to+\infty$ outside a set $E$ of finite $h$-measure, uniformly in $y\in\mathbb{R}$. The conjecture proposes a substantial weakening of the sufficient condition for finite $h$-measure exceptional sets when the density defining the $h$-measure is non-increasing; its resolution is not supplied in the source.