Methods to determine whether a given set is the sum of other sets - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:19:17Z http://mathoverflow.net/feeds/question/56749 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56749/methods-to-determine-whether-a-given-set-is-the-sum-of-other-sets Methods to determine whether a given set is the sum of other sets Stanley Yao Xiao 2011-02-26T16:47:47Z 2011-02-26T19:57:54Z <p>In Tao and Vu's Additive Combinatorics, it is mentioned that "if we replace the additive set $A$ by a larger set, such as $A+B$, $A+A+A$, or $2A - 2A$, then one can locate significantly larger progressions inside these sets..." Thus it would see quite interesting to investigate under what circumstances a given set is of the forms above.</p> <p>On one note, there has been some work done to investigate what $B$ could be if $A, A+B$ are known, namely the complementary base problem. In particular, Vu proved in 2002 that if $\mathcal{P}$ is the set of primes, then there exists a set $B$ such that $|B \cap [1,n]| = O(\log n)$ and the set {$p + b : p \in \mathcal{P}, b \in B$} $= \mathbb{N} \setminus C$, where $C$ is a finite set. </p> <p>My question is, if we know about a set $S$, are there any methods to detect if it contains or is equal to a set of the form $A+B, A+A+A, 2A-2A$ etc.?</p> http://mathoverflow.net/questions/56749/methods-to-determine-whether-a-given-set-is-the-sum-of-other-sets/56763#56763 Answer by Seva for Methods to determine whether a given set is the sum of other sets Seva 2011-02-26T19:57:54Z 2011-02-26T19:57:54Z <p>Almost nothing is known about this, although the question is certainly not new; see Problem 4.11 of <a href="http://math.haifa.ac.il/~seva/Papers/montpr.dvi" rel="nofollow">this joint paper</a> of Ernie Croot and myself.</p>