We had very good stay in the cosy appartement of Vassiliki. The appartement is not very big, but everything you'll need, is there. And, important, it's clean. The terrace and the view at the sea and the beach are wonderful.
We were welcomed by the parents of Vassiliki. When we had questions we could ring Marina, the mother of Vasiliki or sent an e-mail to Vassiliki herself. She always quickly responded.
The sandy beach is clean. The seawater clear. Chairs and umbrella's are available, but it's not commercial.
We recommand restaurant Psili Ammos. It's a nice place that offers good food.
You absolutely need a car or other means of transport, because Pirgaki is a small village and the nearest mini market is about 3 km away. And because Naxos is the largest of the Cycladic Islands a car will bring you at nice places as Agia Anna, Filoti, Apollon, Chalkio, Apiranthos, Chora etc.
We recommand Vasslikis place "Ilia" to everybody who seeks a not touristed and authentic place for staying in Greece/at Naxos.