On a question regarding visibility of lattice points—III

For a positive integer m, let ω(m) denote the number of distinct prime factors of m. Let h(n) be a function defined on the set of positive integers such that h(n)→∞ as n→∞ and let En(h)={d: d is a positive integer, d⩽n, ω(d)⩾h(n)}. Writing Δn={(x,y) : x,y are integers, 1⩽x, y⩽n}, in the present paper we show that one can give explicit description of a set Xn⊂Δn such that Δn is visible from Xn with at most 100|En(h)|2 exceptional points and for all sufficiently large n, one has|Xn|⩽800h(n)log log h(n). As a corollary it follows that one can give explicit description of a set Yn⊂Δn such that for large nʹs, Δn is visible except for at most 100 n2/(log log n)2 exceptional points from Yn where Yn satisfies|Yn|=O((log logn)(log log log log n)).