Floating over the integers
Perl 6, 28 characters



{grep({.Num+|0-$_},^∞)[^$_]}


Try it online!

This is actually 30 bytes, but the challenge is scored in characters.

This should work theoretically. This creates a lazy list of numbers that compares the default number type (with arbitrary precision) against the same number converted to a Num datatype, which is a 64-bit floating point number, and returns the first \$n\$ elements of the list

Here's a version starting at \$2^{53}-5\$