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While there are many open problems in Real Analysis like Khabibullin's conjecture or Lehmer's conjecture, those are big enough to take an expert's life for several years, let alone some graduate student that may haven't had any research done yet.

Are there any problems in Real Analysis which has not been solved yet or otherwise have still some rooms to generalize, on the other hand easy enough to consider it as a project for a research-degree student to finish it in a few years?

Suggestions on this would be very much appreciated!

PS - Migrated from math.SE

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closed as too broad by Bjørn Kjos-Hanssen, Alexandre Eremenko, Alex Degtyarev, Stefan Kohl, Anthony Quas Jun 14 '15 at 12:51

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Picking a research problem from the list of answers you're going to get seems kind of risky. How many other graduate students are going to look at this list and pick the same problem you did? $\endgroup$ – bof Jun 14 '15 at 11:39
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    $\begingroup$ Yes. Many people get PhDs in analysis, so looking at their theses will give you examples. $\endgroup$ – Kimball Jun 14 '15 at 12:45
  • $\begingroup$ @bof In addition, most people won't want to give away their best ideas for problems to grad students who aren't theirs. $\endgroup$ – Kimball Jun 14 '15 at 12:54
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Make Weyl-van der Corput estimates explicit:

Let $f:[a,b]\rightarrow\mathbb{R}$ be a somewhat smooth function, and assume that you have some bounds on certain derivatives of $f$. Then give an upper bound for $\sum_{n=N}^{2N} e^{2\pi i f(n)}$.

There are deep qualitative bounds, however, not much work has been done on the constants involved in these bounds. While it is trivial to get some constants, getting good constants can become quite demanding.

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