The minimum modulus of gap power series and h-measure of exceptional sets

Abstract

For entire Dirichlet series of the form $F(z)=\sum _{n=0}^{+\infty }{a}_n{e}^{z{\lambda }_n},0\le {\lambda }_n\uparrow +\infty (n\to +\infty )$, we establish conditions under which the relation $$F(x+iy)=(1+o(1))a_{\nu (x,F)}{e}^{(x+iy){\lambda }_{\nu (x,F)}}$$ is true as $x\to +\infty$ outside some set $E$ such that $\text{ h-meas }(E)={\int }_{E}dh(x)<+\infty$ uniformly in $y\in \mathbb{R}$, where $h(x)$ is positive continuous function increasing to $+\infty$ on $[0,+\infty )$ with non-decreasing to $+\infty$ derivative.