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**On the critical pair theory in \({\mathbb Z}/p{\mathbb Z}\).**
*(English)*
Zbl 1147.11060

Freĭman’s theorem describes the structure of sets of integers with small doubling, that is sets of integers \(A\) with \(| A+A| = O(| A| )\) where \(A+A:= \{a+a':a,a' \in A\}\). Qualitatively Freĭman’s result is comprehensive, however when the doubling is very small more can be said and, indeed, there is an industry exploring this area across the range of abelian groups.

The paper under review is a contribution to this body. There are considerable differences between small sumset problems in different groups and some of the most striking arise between \(\mathbb Z\) and \(\mathbb Z/p\mathbb Z\): one expects very similar results but the lack of an order on \(\mathbb Z/p\mathbb Z\) renders many proof methods ineffective. Given a subset \(X\) of a cyclic group we write \(l_r(X)\) for the length of the smallest arithmetic progression containing \(X\) – we take the obvious convention that it is infinite if no such exists. In recent collaboration with Rødseth the first author showed the following theorem.

Suppose that \(A,B \subset \mathbb Z/p\mathbb Z\) have \(| A| \geq 3\), \(| B| \geq 4\) and

\[ | A+B| \leq | A| +| B| \leq p-4. \]

Then there is some \(r \in \mathbb Z/p\mathbb Z\) such that \(l_r(A) \leq | A| +1\) and \(l_r(B) \leq | B| +1\).

It does not seem unreasonable to conjecture a generalization of this as follows.

Suppose that \(m \in \mathbb N_0\), \(A,B \subset \mathbb Z/p\mathbb Z\) have \(| A| \geq m+3\) and \(B| \geq m+4\) and

\[ | A+B| \leq | A| +| B| +m \leq p-(m+4). \]

Then there is some \(r \in \mathbb Z/p\mathbb Z\) such that \(l_r(A) \leq | A| +m+1\) and \(l_r(B) \leq | B| +m+1\).

The previous theorem is then the case \(m=0\) of this conjecture and the present paper proves the case \(m=1\) when \(p \geq 53\). The proof uses a variety of ‘one-dimensional isoperimetric tools’ typical in this sort of problem and the paper itself has a very helpful section clearly outlining the method in some detail making it a particularly readable argument.

The paper under review is a contribution to this body. There are considerable differences between small sumset problems in different groups and some of the most striking arise between \(\mathbb Z\) and \(\mathbb Z/p\mathbb Z\): one expects very similar results but the lack of an order on \(\mathbb Z/p\mathbb Z\) renders many proof methods ineffective. Given a subset \(X\) of a cyclic group we write \(l_r(X)\) for the length of the smallest arithmetic progression containing \(X\) – we take the obvious convention that it is infinite if no such exists. In recent collaboration with Rødseth the first author showed the following theorem.

Suppose that \(A,B \subset \mathbb Z/p\mathbb Z\) have \(| A| \geq 3\), \(| B| \geq 4\) and

\[ | A+B| \leq | A| +| B| \leq p-4. \]

Then there is some \(r \in \mathbb Z/p\mathbb Z\) such that \(l_r(A) \leq | A| +1\) and \(l_r(B) \leq | B| +1\).

It does not seem unreasonable to conjecture a generalization of this as follows.

Suppose that \(m \in \mathbb N_0\), \(A,B \subset \mathbb Z/p\mathbb Z\) have \(| A| \geq m+3\) and \(B| \geq m+4\) and

\[ | A+B| \leq | A| +| B| +m \leq p-(m+4). \]

Then there is some \(r \in \mathbb Z/p\mathbb Z\) such that \(l_r(A) \leq | A| +m+1\) and \(l_r(B) \leq | B| +m+1\).

The previous theorem is then the case \(m=0\) of this conjecture and the present paper proves the case \(m=1\) when \(p \geq 53\). The proof uses a variety of ‘one-dimensional isoperimetric tools’ typical in this sort of problem and the paper itself has a very helpful section clearly outlining the method in some detail making it a particularly readable argument.

Reviewer: Tom Sanders (Cambridge)