| Summary: | <p>This thesis is concerned with some problems regarding set addition.</p> <p>One of them is trying to find the correct dependence of parameters in the Balog-Szemerédi-Gowers Theorem. Although we do not resolve this issue completely, we provide a nontrivial lower bound.</p> <p>In the main part of the thesis we develop the structure theory of sets with small popular doubling. We prove various results for subsets of integers, groups of prime order or arbitrary abelian groups; all of them are direct analogues of the results about sets of small doubling.</p> <p>In addition, we provide an estimate of the number of subsets of an interval with certain size and bounded doubling constant; this extends a theorem of Green and Morris. Roughly speaking, we prove that it is enough to count sets that, except a few points, form a dense subset of a (one-dimensional) arithmetic progression. As a tool we prove a result about approximating sets, which might be considered interesting on its own right.</p> <p>Finally, we apply results from previous chapters to strengthen another theorem of Green and Morris. We prove that if a sumset of a random subset of the set of natural numbers misses at least <em>k</em> elements of &Nopf;, with positive probability it means that in fact among those elements we can find all of 1; 2;...,<em>k</em>.</p>
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