The minimal growth of entire functions with given zeros along unbounded sets - I. V. Andrusyak; P.V. Filevych (Математичні Студії, 2020)
Let $l$ be a continuous function on $\mathbb{R}$ increasing to $+∞$, and $φ$ be a positive function on $\mathbb{R}$. We proved that the condition $$ \varliminf_{x→+∞}\frac{φ(\ln[x])}{\ln x}>0 $$ is necessary and sufficient in order that for any complex sequence $(ζ_n)$ with $n(r)\ge l(r)$, $r\ge r_0$, and every set $E\subset\mathbb{R}$ which is unbounded from above there exists an entire function $f$ having zeros only at the points $ζ_n$ such that $$ \varliminf_{r\in E,\ r→+∞}\frac{\ln\ln M_f(r)}{φ(\ln n_ζ(r))\ln l^{-1}(n_ζ(r))}=0. $$ Here $n(r)$ is the counting function of $(ζ_n)$, and $M_f(r)$ is the maximum modulus of $f$.