Things do not exist. To assert any sort of existence is to assert phenomena, or a single phenomenon, to be uncompounded, or unconditioned; causeless.
Neither do things NOT exist.

To say, or to posit or assert, that there is "nothing," is the nihilist position.
We can say, in keeping with convention and the normative experience of sentient beings, only that there is the appearance of things. The Sentient Being experiences the appearance of things. It is the mind, the mental consciousness, which posits "existence."
In fact, the whole idea of "existence" is a conceptual overlay, created by mind. The true nature of "things" is beyond the realm of conceptual understanding, for Buddhists. I believe that can safely be said, without offending any particular doctrinal position.
The question of "existence" is only helpful, to Buddhists, inasmuch as it is a starting point for investigation of ourselves, our minds, and our place in the Universe, for lack of a better word. The goal of Buddha Dharma is not to posit an ontological position, but to liberate sentient beings from suffering. Suffering is caused due to ignorance of the True Nature of things.