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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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Assuming that there exist (infinitely many) Siegel zeros, we show that the (Rosser-)Jurkat-Richert bounds in the linear sieve cannot be improved, and similarly look at Iwaniec's lower bound on Jacobsthal's problem, as well as minor improvements to the Brun-Titchmarsh Theorem. We also deduce an improved (though conditional) lower bound on the longest gaps between primes, and rework Cram\'er's heuristic in this situation to show that we would expect gaps around \( x\) that are significantly larger than \( (\log x)^2.\)

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