Freiman structure and small doubling

A generalized arithmetic progression is obtained by choosing a base point, a finite list of steps, and a positive integer length for each step. Let each coefficient range over its prescribed interval of consecutive integers and add the resulting step contributions to the base point. The rank counts the listed step directions, while the volume is the product of the coefficient lengths. Different coefficient choices can describe the same output. A progression is proper precisely when no such collisions occur within the original coefficient box.

For a proper rank-r progression, its cardinality equals its volume. Adding it to itself gives an image of a box with side lengths twice the original lengths minus one. The enlarged box may fail to be injective even if the original one was proper, so its volume is an upper bound for the sumset cardinality, not automatically an equality. This gives doubling at most two to the rank. It explains bounded doubling for bounded-rank proper progressions without requiring the steps to be linearly independent over the integers.

The qualitative integer theorem says that for every K there are constants depending only on K such that every finite nonempty integer set with doubling at most K lies in a proper generalized progression with rank bounded by one constant and cardinality at most the other constant times the original size. Containment is stronger than merely finding a structured subset, but the containing progression may have holes where the original set has no points. No particular optimal dependence on K is asserted here.

The conclusion does not bound the ordinary diameter by a multiple of the cardinality. An enormous step can separate two short rows of a rank-two progression while leaving its doubling bounded. Nor does the theorem make the set equal to a progression. When BSG has first extracted a high-energy subset, Freiman can be applied to that subset; the final containment then concerns the extracted subset, with constants depending on its inherited doubling. One must not silently enlarge the conclusion back to the original input.

A Freiman homomorphism of order two preserves every relation a+b=c+d among elements of its domain. A Freiman isomorphism is a bijection for which both directions preserve those relations. This is weaker than requiring a group homomorphism on the entire ambient group, yet strong enough to preserve pair-sum collisions and doubling. Such models allow a sparse-looking integer set to be studied in a denser convenient setting, provided the exact relation order needed later is preserved.

In a general abelian group a finite subgroup already has doubling one, regardless of its size or generator requirements. The appropriate structural language therefore permits a finite subgroup factor together with progression directions, called a coset progression. An integer containment theorem should not be transplanted verbatim into torsion settings. Similarly, a proper parameterization must always be checked in the actual ambient arithmetic: modular wraparound may identify coefficient choices that were distinct as integers. These limitations identify what the theorem says rather than weakening its substantial integer conclusion.