The character of Jay Gatsby is full of ambiguities because he is a self-invented character that does not really exist. Jay Gatsby is, like so much else in this story, an illusion that seems to have no substance and when he dies, as is demonstrated by the number of people who attend his funeral, the illusion has disappeared as if it were a bubble in the air, leaving very little trace of its passing. The biggest ambiguity in his character comes through the way that he is presented as the novel develops as a rather naive man full of hope who builds his entire identity around his dreams of being with Daisy. He fails to realise that his dreams just simply aren't worth it. Gatsby, for all of his business acument and ability to gain wealth, does not see that Daisy is a very unworthy receptacle for all of his hopes and desires, and because he heaps so much upon her, he cannot see her weaknesses and her character flaws. A deliberate parallel is constructed between Gatsby's pursuit of Daisy and the American Dream, and this is a key theme in the novel. Note how Nick describes Gatsby's fixated focus on Daisy:
Gatsby believed in the green light, the orgastic future that year by year recedes before us. It eluded us then, but that’s no matter—tomorrow we will run faster, stretch out our arms farther... And then one fine morning—
So we beat on, boats against the current, borne back ceaselessly into the past
The ambiguity surrounding Gatsby's character and surrounding America as a whole in its pursuit of the American Dream is presented through the image of a rower desperately trying to reach the green light ahead of them whilst failing to realise that they are being swept further and further away from it. Gatsby tries to ultimately recreate a moment in history, and fails. Dreams sometimes fail to materialise into reality.