Lacunary tangential approximation

Our aim is to give lacunary versions with upper density equal to the value one of Arakelian's approximation theorem for special geometries of the domain and the closed set.By use of the theorems, we are able to construct functions which have so-called maximal cluster sets along arbitrary curves and which have a lacunary structure in addition. Upper density equal to the value one will turn out to be best possible for this result.