6.3 Applications to structure theory of set addition

Freiman's theorem is a game-changer in additive combinatorics. It tells us that sets with small doubling are basically dense parts of low-dimensional arithmetic progressions. This simple idea has huge implications for understanding the structure of sets in abelian groups.

The theorem connects to key concepts like generalized arithmetic progressions and doubling constants. It's closely linked to the Plünnecke-Ruzsa inequalities and has wide-ranging applications in number theory and combinatorics. Understanding Freiman's theorem opens doors to solving many related problems.

Freiman's theorem for structural properties

Characterizing sets with small doubling

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• Freiman's theorem states that if a finite set $A$ in an abelian group has small doubling, then $A$ is contained in a generalized arithmetic progression (GAP) of bounded dimension
• The doubling constant of a set $A$ is defined as $|A + A| / |A|$, where $A + A = \{a + b : a, b \in A\}$
  • A set has small doubling if its doubling constant is close to 1 (e.g., 1.5)
• GAPs are sets of the form $\{a_0 + k_1a_1 + ... + k_da_d : 0 \leq k_i < n_i\}$, where $a_0, a_1, ..., a_d$ are elements of the abelian group and $n_1, ..., n_d$ are positive integers
  • The dimension of a GAP is $d$
  • Example: $\{1, 3, 5, 7\}$ is a GAP of dimension 1 with $a_0 = 1$, $a_1 = 2$, and $n_1 = 4$

Bounds on the size and dimension of GAPs

• The size of the GAP containing $A$ is bounded by a function of the doubling constant and the size of $A$
  • This function is polynomial in $|A|$ when the doubling constant is fixed
• Freiman's theorem provides a strong structural characterization of sets with small doubling, showing that they are essentially dense subsets of low-dimensional arithmetic progressions
  • For example, if $A$ has doubling constant 1.5, then $A$ is contained in a GAP of dimension at most 3 and size at most $|A|^3$
• The bounds on the size and dimension of the GAP depend on the specific version of Freiman's theorem being used (e.g., Freiman's original theorem, Ruzsa's generalization, or the polynomial Freiman-Ruzsa conjecture)

Freiman's theorem vs Plünnecke-Ruzsa inequalities

Plünnecke-Ruzsa inequalities

• The Plünnecke-Ruzsa inequalities relate the cardinalities of iterated sumsets $A$, $A+A$, $A+A+A$, etc., providing upper bounds on the growth of these sumsets
• For a finite set $A$ in an abelian group, the Plünnecke-Ruzsa inequalities state that $|kA| \leq K^k |A|$ for all positive integers $k$, where $K$ is the doubling constant of $A$ and $kA$ denotes the $k$-fold sumset $A + ... + A$
  • Example: If $A$ has doubling constant 2, then $|3A| \leq 2^3 |A| = 8|A|$

Connections between Freiman's theorem and Plünnecke-Ruzsa inequalities

• Freiman's theorem and the Plünnecke-Ruzsa inequalities are closely related, as both provide information about the structure and growth of sets with small doubling
• The Plünnecke-Ruzsa inequalities can be used to derive bounds on the dimension and size of the GAP containing $A$ in Freiman's theorem
  • For example, if $A$ has doubling constant $K$, then the Plünnecke-Ruzsa inequalities imply that $A$ is contained in a GAP of dimension at most $\log_2 K$ and size at most $K^{\log_2 K} |A|$
• Conversely, Freiman's theorem can be used to prove the Plünnecke-Ruzsa inequalities, demonstrating the deep connection between these two fundamental results in additive combinatorics

Applications of Freiman's theorem

Additive number theory

• Freiman's theorem has numerous applications in additive number theory, particularly in the study of sum-free sets, Sidon sets, and bases for finite abelian groups
• A set $A$ is sum-free if there are no solutions to the equation $a_1 + a_2 = a_3$ with $a_1, a_2, a_3 \in A$
  • Freiman's theorem can be used to bound the maximum size of a sum-free subset of $\{1, 2, ..., n\}$
  • For example, it can be shown that any sum-free subset of $\{1, 2, ..., n\}$ has size at most $n/2 + o(n)$
• A set $A$ is a Sidon set if all the sums $a_1 + a_2$ with $a_1, a_2 \in A$ are distinct
  • Freiman's theorem can be applied to prove upper bounds on the size of Sidon sets in various settings
  • For example, it can be shown that any Sidon set in $\{1, 2, ..., n\}$ has size at most $\sqrt{n} + o(\sqrt{n})$
• In the context of bases for finite abelian groups, Freiman's theorem can be used to characterize the structure of sets $A$ such that every element of the group can be represented as a sum of a bounded number of elements from $A$

Combinatorics

• Freiman's theorem also has applications in combinatorics, particularly in the study of set systems with restricted intersections and in problems involving the additive structure of dense subsets of finite abelian groups
• For example, Freiman's theorem can be used to prove the Erdős-Heilbronn conjecture, which states that any subset of $\mathbb{Z}_p$ (the integers modulo a prime $p$) of size at least $\sqrt{4p}$ contains a three-term arithmetic progression
• Many problems in additive combinatorics and related areas can be reduced to understanding the structure of sets with small doubling, making Freiman's theorem a powerful tool for tackling a wide range of questions in these fields
  • Examples include the study of the clique number of Cayley graphs, the existence of long arithmetic progressions in sumsets, and the structure of sets with small sum set