In additive combinatorics, the Plünnecke-Ruzsa inequality is an inequality that bounds the size of various sumsets of a set, given that there is another set so that is not much larger than. A slightly weaker version of this inequality was originally proven and published by Helmut Plünnecke. Imre Ruzsa later published a simpler proof of the current, more general, version of the inequality. The inequality forms a crucial step in the proof of Freiman's theorem.
The most commonly cited version of the statement of the Plünnecke-Ruzsa inequality is the following. This is often used when, in which case the constant is known as the doubling constant of. In this case, the Plünnecke-Ruzsa inequality states that sumsets formed from a set with small doubling constant must also be small.
Plünnecke's inequality
The version of this inequality that was originally proven by Plünnecke is slightly weaker.
The Ruzsa triangle inequality is an important tool which is used to generalize Plünnecke's inequality to the Plünnecke-Ruzsa inequality. Its statement is:
Proof of Plünnecke-Ruzsa inequality
The following simple proof of the Plünnecke-Ruzsa inequality is due to Petridis. Lemma: Let and be finite subsets of an abelian group. If is a nonempty subset that minimizes the value of, then for all finite subsets, Proof: This is demonstrated by induction on the size of. For the base case of, note that is simply a translation of for any, so For the inductive step, assume the inequality holds for all with for some positive integer. Let be a subset of with, and let for some. Finally, let. The definition of implies that. Thus, by the definition of these sets, Hence, considering the sizes of the sets, The definition of implies that, so by the definition of,. Thus, applying the inductive hypothesis on and using the definition of, To bound the right side of this inequality, let. Suppose and, then there exists such that. Thus, by definition,, so. Hence, the sets and are disjoint. The definitions of and thus imply that Again by definition,, so. Hence, Putting the above two inequalities together gives This completes the proof of the lemma. To prove the Plünnecke-Ruzsa inequality, take and as in the statement of the lemma. It is first necessary to show that This can be proved by induction. For the base case, the definitions of and imply that. Thus, the definition of implies that. For inductive step, suppose this is true for. Applying the lemma with and the inductive hypothesis gives This completes the induction. Finally, the Ruzsa triangle inequality gives Because, it must be the case that. Therefore, This completes the proof of the Plünnecke-Ruzsa inequality.
Both Plünnecke's proof of Plünnecke's inequality and Ruzsa's original proof of the Plünnecke-Ruzsa inequality use the method of Plünnecke graphs. Plünnecke graphs are a way to capture the additive structure of the sets in a graph theoretic manner First important to defining Plünnecke graphs is the notion of a commutative graph. Definition. A directed graph is called semicommutative if, whenever there exist distinct such that and are edges in for each, then there also exist distinct so that and are edges in for each. is called commutative if it is semicommutative and the graph formed by reversing all its edges is also semicommutative. Now a Plünnecke graph is defined as follows. Definition. A layered graph is a graph whose vertex set can be partitioned so that all edges in are from to, for some. A Plünnecke graph is a layered graph which is commutative. The relevant example of a Plünnecke graph is the following, showing how the structure of the sets is a case of that of a Plünnecke graph. Example. Let be subsets of an abelian group. Then, let be the layered graph so that each layer is a copy of, so that,,..., . Create the edge whenever there exists such that. Then is a Plünnecke graph. For example, to check that is semicommutative, if and are edges in for each, then. Then, let, so that and. Thus, is semicommutative. It can be similarly checked that the graph formed by reversing all edges of is also semicommutative, so is a Plünnecke graph. In a Plünnecke graph, the image of a set in, written, is defined to be the set of vertices in which can be reached by a path starting from some vertex in. In particular, in the aforementioned example, is just. The magnification ratio between and, denoted, is then defined as the minimum factor by which the image of a set must exceed the size of the original set. Formally, Plünnecke's theorem is the following statement about Plünnecke graphs. The proof of Plünnecke's theorem involves a technique known as the "tensor product trick", in addition to an application of Menger's theorem. The Plünnecke-Ruzsa inequality is a fairly direct consequence of Plünnecke's theorem and the Ruzsa triangle inequality. Applying Plünnecke's theorem to the graph given in the example, at and, yields that if, then there exists so that. Applying this result once again with instead of, there exists so that. Then, by Ruzsa's triangle inequality, thus proving the Plünnecke-Ruzsa inequality.
