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Preprints submitted for publication

Let \( A = \{0 = a_0 < a_1 < \cdots < a_{\ell + 1} = b\} \) be a finite set of non-negative integers. We prove that the sumset \( NA \) has a certain easily-described structure, provided that \( N \geqslant b-\ell \), as recently conjectured by Granville and Shakan. We also classify those sets \( A \) for which this bound cannot be improved.

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We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length \( y\) around \( x\), where \( y\ll (\log x)^2\). In particular we conjecture that the maximum grows surprisingly slowly as \( y\) ranges from \( \log x\) to \( (\log x)^2\). We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification.

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