Did you know that e^pi - pi is ALMOST an integer? It is 19.999099979... . For a fun diversion in 'almost integers' seehttps://bit.ly/2RIgIb0
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By looking through the link and Wolfram mathworld, they seem to work with an informal definition. One way you could define it would be "x is an almost integer of error eps if there is an integer y such that |x-y|<eps". I think you would need the error part to make it well defined
Nice.. would there then be well defined notion of convergence to an integer?
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Sure you could just use the definition of convergence. This almost integer thing is that we are using combinations of irrationals to approximate integers, so convergence of approximations would be a natural next step.
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One way you could construct a "almost integer sequence" is find a function whose stable fixed point is an integer and start a fixed point iteration with an irrational if you want.
End of conversation
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