EDIT: A small change in hypothesis like, changing the space from Zygmund class of functions to Log-lipschitz class or looking for an integer solution in a equation where the solution is easy to find in real or complex case. For instance, Fermat's last theorem is easy to solve on real-line or complex-plane but it engaged mathematicians for three centuries in the integer case. (I have just turned around the example in comment by B. Bischof given below and it is suitable here!)
I thought, and still think, that changing from "asking for real solutions" to "asking for integer solutions" is neither a small change nor a loose hypothesis, but a drastic and dramatic change of the problem.
My misgivings are that the question seemed to invite somewhat superficial answers, and would attract more in the future, getting bumped up to the top each time someone had an idea. (Changing from $L^p$ to $L^1$ is not a small change, it is a big change, and not so much a case of a loose hypothesis as a cavalier approach to precision.)
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