# Near optimal bounds in Freiman's theorem

@article{Schoen2011NearOB, title={Near optimal bounds in Freiman's theorem}, author={Tomasz Schoen}, journal={Duke Mathematical Journal}, year={2011}, volume={158}, pages={1-12} }

We prove that if for a finite set A of integers we have |A+A| ≤ K|A|, then A is contained in a generalized arithmetic progression of dimension at most K −1/2 and size at most exp(K −1/2 )|A| for some absolute constant C. We also discuss a number of applications of this result.

## 45 Citations

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## References

SHOWING 1-10 OF 31 REFERENCES

Freiman's theorem in an arbitrary abelian group

- Mathematics
- 2005

A famous result of Freiman describes the structure of finite sets A ⊆ ℤ with small doubling property. If |A + A| ⩽ K|A|, then A is contained within a multidimensional arithmetic progression of…

Finite and infinite arithmetic progressions in sumsets

- Mathematics
- 2006

We prove that if A is a subset of at least cn1/2 elements of {1, . . . , n}, where c is a sufficiently large constant, then the collection of subset sums of A contains an arithmetic progression of…

A new proof of Szemerédi's theorem

- Mathematics
- 2001

In 1927 van der Waerden published his celebrated theorem on arithmetic progressions, which states that if the positive integers are partitioned into finitely many classes, then at least one of these…

Appendix to ‘Roth’s theorem on progressions revisited,’ by J. Bourgain

- Mathematics
- 2007

We show two results. First, a refinement of Freiman's theorem: if A is a finite set of integers and |A+A| > |A|(log |A|)^{4/3-\epsilon} for all \epsilon>0.

Finite field models in additive combinatories

- MathematicsBCC
- 2005

The study of many problems in additive combinatorics, such as Szemer\'edi's theorem on arithmetic progressions, is made easier by first studying models for the problem in F_p^n for some fixed small…

On Additive Doubling and Energy

- Mathematics, Computer ScienceSIAM J. Discret. Math.
- 2010

It is shown that there is a universal $\epsilon>0$ so that any subset of an abelian group with subtractive doubling $K$ must be polynomially related to a set with additive energy at least $\frac{1}{K^{1-\ep silon}}$.

A polynomial bound in Freiman's theorem

- Mathematics
- 2002

.Earlier bounds involved exponential dependence in αin the second estimate. Ourargument combines I. Ruzsa’s method, which we improve in several places, as well asY. Bilu’s proof of Freiman’s…

Appendix to ‘Roth’s theorem on progressions revisited,’ by

- Mathematics
- 2008

The previous best estimates are due to Chang [Cha02] who showed the above result (up to logarithmic factors) with 2 in place of 7/4. Note that one cannot hope to improve the dimension bound past ⌊K −…