Intermediate and small scale limiting theorems for random fields

In this paper we study the nodal lines of random eigenfunctions Laplacian on torus, so called 'arithmetic waves'. To be more precise, number intersections line with a straight interval in given direction. We are interested how depends length and direction distribution spectral measure wave. analyse second factorial moment short regime persistence probability long regime. also relations between Cilleruelo Cilleruelo-type fields. give an explicit coupling these fields which mesoscopic scales preserves structure sets close to one.