The Typical Structure of Small Sumsets ∗ M. Wötzel 1 M. Campos M. Coulson G. Perarnau O. Serra 1 Universitat 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 | 2 𝐵 | ≤ 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 | 2 𝐵 | ≤ 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 2 𝐵 ⊂ 2 𝑄 and hence | 2 𝐵 | ≤ | 2 𝑄 | ≈ 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 | 2 𝐵 | ≤ 𝐿𝑡 there exists an arithmetic progression 𝑄 of size | 𝑄 | ≤ 1 + 𝑝 ( 1 ) 𝐿𝑡 2 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 | 2 𝐵 | ≤ 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 ( 𝐿 2 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 | 2 𝐵 | ≤ 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 ( 𝐿 2 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 ( 𝐿 2 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 ) 𝐿𝑡 2 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 ⊂ 2 𝑊 (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 ⊂ 2 𝑊 (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 ⊂ 2 𝑊 (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 of 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 of 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 𝐿 𝑜 ). 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 of 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 𝐿 𝑜 ). 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 of 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 𝐿 𝑜 ). 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
Recommend
More recommend
