Classical Inverse Results and an Observation Theorem (Folklore) - - PowerPoint PPT Presentation

The Typical Structure of Small Sumsets∗

• M. Campos
• M. Coulson
• G. Perarnau
• O. Serra
• M. Wötzel1

1Universitat Politècnica de Catalunya & Barcelona Graduate School of Mathematics

Additive Combinatorics in Marseille 2020 September 9, 2020

∗Thanks to support from MDM-2014-0445.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 1 / 17

Classical Inverse Results and an Observation

Theorem (Folklore)

Suppose 𝐵, 𝐶 ⊂ Z are finite. Then |𝐵 + 𝐶| = |𝐵| + |𝐶| − 1, if and only if 𝐵 and 𝐶 are arithmetic progressions with the same common difference.

Theorem (Freiman)

If 𝐵 ⊂ Z is finite such that |2𝐵| ≤ 𝐿|𝐵|, then 𝐵 is contained in a generalized arithmetic progression of dimension 𝑒(𝐿) and size 𝑔 (𝐿)|𝐵|.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 2 / 17

Classical Inverse Results and an Observation

Theorem (Folklore)

Suppose 𝐵, 𝐶 ⊂ Z are finite. Then |𝐵 + 𝐶| = |𝐵| + |𝐶| − 1, if and only if 𝐵 and 𝐶 are arithmetic progressions with the same common difference.

Theorem (Freiman)

If 𝐵 ⊂ Z is finite such that |2𝐵| ≤ 𝐿|𝐵|, then 𝐵 is contained in a generalized arithmetic progression of dimension 𝑒(𝐿) and size 𝑔 (𝐿)|𝐵|. Suppose that 𝑄 ⊂ Z is an arithmetic progression of size 𝐿𝑡/2. If 𝐵 ⊂ 𝑄 is an arbitrary subset of size 𝑡, then we clearly have 2𝐵 ⊂ 2𝑄 and hence |2𝐵| ≤ |2𝑄| ≈ 𝐿𝑡. Question: Can we get an inverse result in this direction by going to the random setting?

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 2 / 17

The Typical Case - Rough Structure

Question: Can we get an inverse result in this direction by going to the random setting? Answer: Yes, Campos proved the following rough structural result.

Theorem (Campos)

Let 𝑡 = Ω((log 𝑜)3) and 𝐿 = 𝑃(𝑡/(log 𝑜)3).

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 3 / 17

The Typical Case - Rough Structure

Question: Can we get an inverse result in this direction by going to the random setting? Answer: Yes, Campos proved the following rough structural result.

Theorem (Campos)

Let 𝑡 = Ω((log 𝑜)3) and 𝐿 = 𝑃(𝑡/(log 𝑜)3). Then for almost all sets 𝐵 ⊂ [𝑜] with |𝐵| = 𝑡 and |2𝐵| ≤ 𝐿𝑡 there exists an arithmetic progression 𝑄 of size |𝑄| ≤ 1 + 𝑝(1) 2 𝐿𝑡 such that at most 𝑝(𝑡) points of 𝐵 are not contained in 𝑄. Note: Campos also proved a counting result about the number of 𝑡-sets with doubling constant 𝐿.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 3 / 17

The Typical Case - Precise Structure

This was then used to get the following more precise structural result in the case 𝐿 = 𝑃(1).

Theorem (Campos, Collares, Morris, Morrison, Souza)

Fix 𝐿 ≥ 3 and 𝜗 > 0. For 𝑜 sufficiently large, let 𝑡 ≥ (log 𝑜)4.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 4 / 17

The Typical Case - Precise Structure

This was then used to get the following more precise structural result in the case 𝐿 = 𝑃(1).

Theorem (Campos, Collares, Morris, Morrison, Souza)

Fix 𝐿 ≥ 3 and 𝜗 > 0. For 𝑜 sufficiently large, let 𝑡 ≥ (log 𝑜)4. Then for all but an 𝜗 proportion of sets 𝐵 ⊂ [𝑜] with |𝐵| = 𝑡 and |2𝐵| ≤ 𝐿𝑡, it holds that 𝐵 is contained in an arithmetic progression 𝑄 of size |𝑄| ≤ 𝐿𝑡 2 + 𝑑(𝐿, 𝜗), where 𝑑(𝐿, 𝜗) = 𝑃(𝐿2 log 𝐿 log(1/𝜗)).

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 4 / 17

The Typical Case - Precise Structure

This was then used to get the following more precise structural result in the case 𝐿 = 𝑃(1).

Theorem (Campos, Collares, Morris, Morrison, Souza)

Fix 𝐿 ≥ 3 and 𝜗 > 0. For 𝑜 sufficiently large, let 𝑡 ≥ (log 𝑜)4. Then for all but an 𝜗 proportion of sets 𝐵 ⊂ [𝑜] with |𝐵| = 𝑡 and |2𝐵| ≤ 𝐿𝑡, it holds that 𝐵 is contained in an arithmetic progression 𝑄 of size |𝑄| ≤ 𝐿𝑡 2 + 𝑑(𝐿, 𝜗), where 𝑑(𝐿, 𝜗) = 𝑃(𝐿2 log 𝐿 log(1/𝜗)). Note that the value of 𝑑 is close to optimal, as there is also a lower bound of the form Ω(𝐿2 log(1/𝜗)).

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 4 / 17

Our Result

We generalize Campos’ original result to distinct sets.

Theorem (Campos, Coulson, Perarnau, Serra, W.)

Let 𝑡 = Ω((log 𝑜)3) and 𝐿 = 𝑃(𝑡/(log 𝑜)3).

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 5 / 17

Our Result

We generalize Campos’ original result to distinct sets.

Theorem (Campos, Coulson, Perarnau, Serra, W.)

Let 𝑡 = Ω((log 𝑜)3) and 𝐿 = 𝑃(𝑡/(log 𝑜)3). Then for almost all sets 𝐵, 𝐶 ⊂ [𝑜] with |𝐵| = |𝐶| = 𝑡 and |𝐵 + 𝐶| ≤ 𝐿𝑡 there exist arithmetic progressions 𝑄, 𝑅 with the same common difference of size |𝑄|, |𝑅| ≤ 1 + 𝑝(1) 2 𝐿𝑡 such that |𝐵 \ 𝑄|, |𝐶 \ 𝑅| = 𝑝(𝑡). More precise bounds for the 𝑝 terms are obtained, similar but slightly weaker to those of Campos. Main tool in the proof: recent version of the method of hypergraph containers.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 5 / 17

The Method of Hypergraph Containers

Introduced explicitly in 2015 by Balogh, Morris and Samotij, and independently by Saxton and Thomason, used to count the number of combinatorial objects avoiding some specific substructure.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 6 / 17

The Method of Hypergraph Containers

Introduced explicitly in 2015 by Balogh, Morris and Samotij, and independently by Saxton and Thomason, used to count the number of combinatorial objects avoiding some specific substructure. General idea: If H is a hypergraph satisfying some specific degree conditions, then there exists a relatively small family C ⊂ 2𝑊 (H) of containers such that:

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 6 / 17

The Method of Hypergraph Containers

Introduced explicitly in 2015 by Balogh, Morris and Samotij, and independently by Saxton and Thomason, used to count the number of combinatorial objects avoiding some specific substructure. General idea: If H is a hypergraph satisfying some specific degree conditions, then there exists a relatively small family C ⊂ 2𝑊 (H) of containers such that:

1 For each independent set 𝐽 ⊂ 𝑊(H), there exists a container 𝐷 ∈ C with

𝐽 ⊂ 𝐷.

2 Each 𝐷 is smaller than 𝑊(H) by some constant factor.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 6 / 17

The Method of Hypergraph Containers

Introduced explicitly in 2015 by Balogh, Morris and Samotij, and independently by Saxton and Thomason, used to count the number of combinatorial objects avoiding some specific substructure. General idea: If H is a hypergraph satisfying some specific degree conditions, then there exists a relatively small family C ⊂ 2𝑊 (H) of containers such that:

1 For each independent set 𝐽 ⊂ 𝑊(H), there exists a container 𝐷 ∈ C with

𝐽 ⊂ 𝐷.

2 Each 𝐷 is smaller than 𝑊(H) by some constant factor.

As long as the degree conditions are met, one can now reapply this result to the induced hypergraph on each container 𝐷 ∈ C.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 6 / 17

The Method of Hypergraph Containers

After iterating, end up with a still small-ish collection of containers, each

• f which is very small, and it still holds that every independent set 𝐽 of H

is contained in one of these containers.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 7 / 17

The Method of Hypergraph Containers

After iterating, end up with a still small-ish collection of containers, each

• f which is very small, and it still holds that every independent set 𝐽 of H

is contained in one of these containers. Consider now a specific H whose hyperedges encode some structure (e.g. triangles in 𝐿𝑜).

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 7 / 17

The Method of Hypergraph Containers

After iterating, end up with a still small-ish collection of containers, each

• f which is very small, and it still holds that every independent set 𝐽 of H

is contained in one of these containers. Consider now a specific H whose hyperedges encode some structure (e.g. triangles in 𝐿𝑜). The iteration stopped, hence the induced hypergraph on each container has few edges (i.e., few triangles).

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 7 / 17

The Method of Hypergraph Containers

After iterating, end up with a still small-ish collection of containers, each

• f which is very small, and it still holds that every independent set 𝐽 of H

is contained in one of these containers. Consider now a specific H whose hyperedges encode some structure (e.g. triangles in 𝐿𝑜). The iteration stopped, hence the induced hypergraph on each container has few edges (i.e., few triangles). Can now use classical stability results to get structural statements about the containers, which translate down to the independent sets (i.e. triangle free graphs).

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 7 / 17

An Asymmetric Version of the Container Lemma

Original method works well when e.g. counting the number or determining structure of 𝐼-free graphs, for some fixed graph 𝐼.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 8 / 17

An Asymmetric Version of the Container Lemma

Original method works well when e.g. counting the number or determining structure of 𝐼-free graphs, for some fixed graph 𝐼. Problem: When trying to get results about induced subgraphs, one wants to consider both edges and non-edges.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 8 / 17

An Asymmetric Version of the Container Lemma

Original method works well when e.g. counting the number or determining structure of 𝐼-free graphs, for some fixed graph 𝐼. Problem: When trying to get results about induced subgraphs, one wants to consider both edges and non-edges. In order to give a structure theorem about induced-𝐷4-free graphs, Morris, Samotij and Saxton recently developed an asymmetric (bipartite) version of the container method.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 8 / 17

An Asymmetric Version of the Container Lemma

Original method works well when e.g. counting the number or determining structure of 𝐼-free graphs, for some fixed graph 𝐼. Problem: When trying to get results about induced subgraphs, one wants to consider both edges and non-edges. In order to give a structure theorem about induced-𝐷4-free graphs, Morris, Samotij and Saxton recently developed an asymmetric (bipartite) version of the container method. Campos slightly modified this so that the two components can shrink at different rates and allowed non-uniform hypergraphs.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 8 / 17

Proof Outline – Generalizing the Asymmetric Container Lemma

We generalize the bipartite container lemma to an 𝑠-partite, 𝑠-bounded version and apply it to the following hypergraph.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 9 / 17

Proof Outline – Generalizing the Asymmetric Container Lemma

We generalize the bipartite container lemma to an 𝑠-partite, 𝑠-bounded version and apply it to the following hypergraph. For (𝑊1,𝑊2,𝑊3) ∈ 2[𝑜] × 2[𝑜] × 2[2𝑜], we consider the 3-partite and 3-uniform hypergraph H (𝑊1,𝑊2,𝑊3) with vertex set 𝑊1 ∪ 𝑊2 ∪ 𝑊3 and hyperedges {(𝑦, 𝑧, 𝑨) ∈ 𝑊1 × 𝑊2 × 𝑊3 : 𝑦 + 𝑧 = 𝑨}.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 9 / 17

Proof Outline – Generalizing the Asymmetric Container Lemma

We generalize the bipartite container lemma to an 𝑠-partite, 𝑠-bounded version and apply it to the following hypergraph. For (𝑊1,𝑊2,𝑊3) ∈ 2[𝑜] × 2[𝑜] × 2[2𝑜], we consider the 3-partite and 3-uniform hypergraph H (𝑊1,𝑊2,𝑊3) with vertex set 𝑊1 ∪ 𝑊2 ∪ 𝑊3 and hyperedges {(𝑦, 𝑧, 𝑨) ∈ 𝑊1 × 𝑊2 × 𝑊3 : 𝑦 + 𝑧 = 𝑨}. Note that the independent sets 𝐽 we are interested in will satisfy the additional condition that they have a large intersection with the third component, more precisely |𝐽 ∩ 𝑊3| ≥ |𝑊3| − 𝐿𝑡.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 9 / 17

Proof Outline – Generalizing the Asymmetric Container Lemma

We generalize the bipartite container lemma to an 𝑠-partite, 𝑠-bounded version and apply it to the following hypergraph. For (𝑊1,𝑊2,𝑊3) ∈ 2[𝑜] × 2[𝑜] × 2[2𝑜], we consider the 3-partite and 3-uniform hypergraph H (𝑊1,𝑊2,𝑊3) with vertex set 𝑊1 ∪ 𝑊2 ∪ 𝑊3 and hyperedges {(𝑦, 𝑧, 𝑨) ∈ 𝑊1 × 𝑊2 × 𝑊3 : 𝑦 + 𝑧 = 𝑨}. Note that the independent sets 𝐽 we are interested in will satisfy the additional condition that they have a large intersection with the third component, more precisely |𝐽 ∩ 𝑊3| ≥ |𝑊3| − 𝐿𝑡. Think: We want to count triples (𝐵, 𝐶, 𝐷) such that |𝐵| = |𝐶| = 𝑡, 𝐵 + 𝐶 ⊂ 𝐷 and |𝐷| ≤ 𝐿𝑡.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 9 / 17

Proof Outline – The Container Theorem

We get the following theorem.

Theorem (CCPOW)

Let 𝑜, 𝑛, 𝑡 be integers such that log 𝑜 ≤ 𝑡 ≤ 𝑛 ≤ 𝑡2 and 𝜗 > 0.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 10 / 17

Proof Outline – The Container Theorem

We get the following theorem.

Theorem (CCPOW)

Let 𝑜, 𝑛, 𝑡 be integers such that log 𝑜 ≤ 𝑡 ≤ 𝑛 ≤ 𝑡2 and 𝜗 > 0. There is a family A ⊂ 2[𝑜] × 2[𝑜] × 2[2𝑜] of triples of sets (𝑌,𝑍, 𝑎) of size |A| ≤ exp(𝑃(√𝑛𝜗−2(log 𝑜)3/2)), such that:

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 10 / 17

Proof Outline – The Container Theorem

We get the following theorem.

Theorem (CCPOW)

Let 𝑜, 𝑛, 𝑡 be integers such that log 𝑜 ≤ 𝑡 ≤ 𝑛 ≤ 𝑡2 and 𝜗 > 0. There is a family A ⊂ 2[𝑜] × 2[𝑜] × 2[2𝑜] of triples of sets (𝑌,𝑍, 𝑎) of size |A| ≤ exp(𝑃(√𝑛𝜗−2(log 𝑜)3/2)), such that:

1 For all 𝐵, 𝐶 ⊂ [𝑜], 𝐷 ⊂ [2𝑜] with |𝐵| = |𝐶| = 𝑡, 𝐵 + 𝐶 ⊂ 𝐷 and |𝐷| ≤ 𝑛,

there is a triple (𝑌,𝑍, 𝑎) ∈ A such that 𝐵 ⊂ 𝑌, 𝐶 ⊂ 𝑍 and 𝑎 ⊂ 𝐷.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 10 / 17

Proof Outline – The Container Theorem

We get the following theorem.

Theorem (CCPOW)

Let 𝑜, 𝑛, 𝑡 be integers such that log 𝑜 ≤ 𝑡 ≤ 𝑛 ≤ 𝑡2 and 𝜗 > 0. There is a family A ⊂ 2[𝑜] × 2[𝑜] × 2[2𝑜] of triples of sets (𝑌,𝑍, 𝑎) of size |A| ≤ exp(𝑃(√𝑛𝜗−2(log 𝑜)3/2)), such that:

1 For all 𝐵, 𝐶 ⊂ [𝑜], 𝐷 ⊂ [2𝑜] with |𝐵| = |𝐶| = 𝑡, 𝐵 + 𝐶 ⊂ 𝐷 and |𝐷| ≤ 𝑛,

there is a triple (𝑌,𝑍, 𝑎) ∈ A such that 𝐵 ⊂ 𝑌, 𝐶 ⊂ 𝑍 and 𝑎 ⊂ 𝐷.

2 For every (𝑌,𝑍, 𝑎) ∈ A, |𝑎| ≤ 𝑛 and either max{|𝑌|, |𝑍|} < 𝑛/log 𝑜 or

there are at most 𝜗2|𝑌||𝑍| pairs (𝑦, 𝑧) ∈ 𝑌 × 𝑍 such that 𝑦 + 𝑧 ∉ 𝑎.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 10 / 17

Proof Outline – A Stability Result for Distinct Sets of Different Sizes

Note that while the sets 𝐵, 𝐶 in the end will have the same cardinality, the containers 𝑌 and 𝑍 might not, so we need a stability result for this more general case.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 11 / 17

Proof Outline – A Stability Result for Distinct Sets of Different Sizes

Note that while the sets 𝐵, 𝐶 in the end will have the same cardinality, the containers 𝑌 and 𝑍 might not, so we need a stability result for this more general case. Following arguments by Lev and Shao and Xu, we prove the following.

Lemma

Let 𝜗 ≤ 2−9. Let 𝑌,𝑍, 𝑎 ⊂ Z such that (1 − 𝜗)|𝑎| ≤ |𝑌| + |𝑍| and max{|𝑌|, |𝑍|} ≤ (1 + 4√𝜗)|𝑎|/2, then one of the following holds:

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 11 / 17

Proof Outline – A Stability Result for Distinct Sets of Different Sizes

Note that while the sets 𝐵, 𝐶 in the end will have the same cardinality, the containers 𝑌 and 𝑍 might not, so we need a stability result for this more general case. Following arguments by Lev and Shao and Xu, we prove the following.

Lemma

Let 𝜗 ≤ 2−9. Let 𝑌,𝑍, 𝑎 ⊂ Z such that (1 − 𝜗)|𝑎| ≤ |𝑌| + |𝑍| and max{|𝑌|, |𝑍|} ≤ (1 + 4√𝜗)|𝑎|/2, then one of the following holds:

1 There are at least 𝜗2|𝑌||𝑍| pairs (𝑦, 𝑧) ∈ 𝑌 × 𝑍 such that 𝑦 + 𝑧 ∉ 𝑎.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 11 / 17

Proof Outline – A Stability Result for Distinct Sets of Different Sizes

Note that while the sets 𝐵, 𝐶 in the end will have the same cardinality, the containers 𝑌 and 𝑍 might not, so we need a stability result for this more general case. Following arguments by Lev and Shao and Xu, we prove the following.

Lemma

Let 𝜗 ≤ 2−9. Let 𝑌,𝑍, 𝑎 ⊂ Z such that (1 − 𝜗)|𝑎| ≤ |𝑌| + |𝑍| and max{|𝑌|, |𝑍|} ≤ (1 + 4√𝜗)|𝑎|/2, then one of the following holds:

1 There are at least 𝜗2|𝑌||𝑍| pairs (𝑦, 𝑧) ∈ 𝑌 × 𝑍 such that 𝑦 + 𝑧 ∉ 𝑎. 2 There are arithmetic progressions 𝑄, 𝑅 of length at most |𝑎|/2 + 3√𝜗|𝑎|

with the same common difference such that 𝑄 contains all but at most 𝜗|𝑌| points of 𝑌, and similarly for 𝑅 and 𝑍. If the second property holds, we say that 𝑌 is (𝜗, |𝑎|)-close to 𝑄.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 11 / 17

Proof Outline – Structure of The Container Family

Applying this stability result to the family A from our container theorem, we see that every container triple (𝑌,𝑍, 𝑎) ∈ A satisfies one of the following properties.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 12 / 17

Proof Outline – Structure of The Container Family

Applying this stability result to the family A from our container theorem, we see that every container triple (𝑌,𝑍, 𝑎) ∈ A satisfies one of the following properties.

1 |𝑌| + |𝑍| ≤ (1 − 𝜗)𝐿𝑡, 2 max{|𝑌|, |𝑍|} > (1 + 4√𝜗)𝐿𝑡/2, or 3 𝑌 and 𝑍 are (𝜗, 𝐿𝑡)-close to arithmetic progressions with the same

common difference.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 12 / 17

Proof Outline – Counting the Exceptions (i)

Suppose the container triple (𝑌,𝑍, 𝑎) ∈ A satisfies either

1 |𝑌| + |𝑍| ≤ (1 − 𝜗)𝐿𝑡, or 2 max{|𝑌|, |𝑍|} > (1 + 4√𝜗)𝐿𝑡/2.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 13 / 17

Proof Outline – Counting the Exceptions (i)

Suppose the container triple (𝑌,𝑍, 𝑎) ∈ A satisfies either

1 |𝑌| + |𝑍| ≤ (1 − 𝜗)𝐿𝑡, or 2 max{|𝑌|, |𝑍|} > (1 + 4√𝜗)𝐿𝑡/2.

Then one can use standard identities and bounds for the binomial coefficient to show that at most 𝑝(1) 𝐿𝑡/2 𝑡 2 pairs of 𝑡-sets 𝐵 ⊂ 𝑌, 𝐶 ⊂ 𝑍 exist.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 13 / 17

Proof Outline – Counting the Exceptions (ii)

Finally, suppose the container triple (𝑌,𝑍, 𝑎) ∈ A satisfies 𝑌 and 𝑍 are (𝜗, 𝐿𝑡)-close to arithmetic progressions 𝑄, 𝑅 with the same common difference.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 14 / 17

Proof Outline – Counting the Exceptions (ii)

Finally, suppose the container triple (𝑌,𝑍, 𝑎) ∈ A satisfies 𝑌 and 𝑍 are (𝜗, 𝐿𝑡)-close to arithmetic progressions 𝑄, 𝑅 with the same common difference. For this case, we exploit the structure of 𝑌 and 𝑍, as well as a general upper bound on the size of their sum to show that there are only 𝑝(1)𝐿 𝑡/2

𝑡

2 pairs of sets 𝐵 ⊂ 𝑌 and 𝐶 ⊂ 𝑍, both of cardinality 𝑡, such that one of them is not (Ω(𝜗), 𝐿𝑡)-close to 𝑄 or 𝑅.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 14 / 17

The Theorem Once More

Using these counting results and the fact that one can take a single pair of disjoint arithmetic progressions 𝑄, 𝑅 with the same common difference of size 𝐿𝑡/2, we arrive again at our main result.

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 15 / 17

The Theorem Once More

Using these counting results and the fact that one can take a single pair of disjoint arithmetic progressions 𝑄, 𝑅 with the same common difference of size 𝐿𝑡/2, we arrive again at our main result.

Theorem (Campos, Coulson, Perarnau, Serra, W.)

Let 𝑡 = Ω((log 𝑜)3) and 𝐿 = 𝑃(𝑡/(log)3). Then for almost all sets 𝐵, 𝐶 ⊂ [𝑜] with |𝐵| = |𝐶| = 𝑡 and |𝐵 + 𝐶| ≤ 𝐿𝑡 there exist arithmetic progressions 𝑄, 𝑅 with the same common difference of size |𝑄|, |𝑅| ≤ 1 + 𝑝(1) 2 𝐿𝑡 such that |𝐵 \ 𝑄|, |𝐶 \ 𝑅| = 𝑝(𝑡).

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 15 / 17

Open Questions

What if 𝐵 and 𝐶 are not of the same size? More than two distinct summands? Groups other than Z? What about the precise structural version of CCMMS?

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 16 / 17

Thank you for your attention!

• M. Wötzel (UPC & BGSMath)

Typical Small Sumsets CAM2020 17 / 17