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A finite set of integers A tiles the integers by translations if Z can be covered by pairwise disjoint translated copies of A. Restricting attention to one tiling period, we have for some N and Z. This can also be stated in terms of cyclotomic divisibility of the mask polynomials associated with A and B. In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids and saturating sets, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a set A containing certain configurations can tile a cyclic group ZM, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in [24] that all tilings of period where are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz [2]. [ABSTRACT FROM AUTHOR]

We prove results about the Lp-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in Lp, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,. . .,N} of sizes αN and βN then A+B contains an arithmetic progression of length at least \begin{equation} \exp ( c (\alpha \beta \log N)^{1/2} - \log\log N). \end{equation} Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least \begin{equation} \exp\biggl( c \biggl(\frac{\alpha \log N}{\log^3 2\beta^{-1}} \biggr)^{1/2} - \log( \beta^{-1} \log N) \biggr). \end{equation} [ABSTRACT FROM AUTHOR]