P N umber T heory Olivier Ramar A dditive C ombinatorics - PowerPoint PPT Presentation

Background Basic Problem The P 2 -path Some examples References S ome ❆❉❉■❚■❱❊ ◗❯❊❙❚■❖◆❙ in ▼❯▲❚■P▲■❈❆❚■❱❊ N umber T heory Olivier Ramaré A dditive C ombinatorics in M arseille 2020 September 6, 2020

Background Basic Problem The P 2 -path Some examples References Background Together with ◮ Aled Walker 2016 ◮ Oriol Serra & Priyamvad Srivastav 2018 ◮ Balasubramanian & Priyamvad Srivastav – in progress. Theorem Let q ≥ 1 and a prime to q ∃ p 1 , p 2 , p 3 ≤ ( 650 q ) 3 / p 1 p 2 p 3 ≡ a [ q ] . (2018) (In progress) q ≥ 10 30 and p 1 , p 2 , p 3 ≤ ( 9000 q ) 11 / 4 .

Background Basic Problem The P 2 -path Some examples References Also in Number Fields Also different groups! Together with Sanoli Gun and Jyothsnaa Sivaraman Theorem Let K be a number field, ∃ c ( K ) / For any q integral ideal, each class of H q ( K ) (the narrow ray class group) contains a p 1 p 2 p 3 ◮ N K / Q p 1 , p 2 , p 3 ≤ c ( K ) N K / Q ( q ) 16 / 3 + ε ◮ p 1 , p 2 , p 3 of degree one. In progress – Dependence on d ( K ) ? To be compared with (Zaman, 2016) (ray class field, exceptional case) and with (Zaman, 2017) (Tchebotarev)

Background Basic Problem The P 2 -path Some examples References Basic Problem Let G be a finite abelian group, Let A ⊂ G be such that |A| / G ≥ η > 1 / 3, Find conditions so that 3 A = G . What kind of conditions?? C onditions : ◮ [ C 0 ] A generates G . ◮ [ C 1 ] A intersects every subgroup of G of index 2 . How do we verify these conditions? Brun-Titchmarsh inequality and its variants . [ p ≤ q α and η = η ( α ) decreases with α ].

Background Basic Problem The P 2 -path Some examples References Special case of Kneser’s Theorem: Lemma Let A ⊂ G (G finite ABELIAN group) . Let H subgroup of G that stabilizes A + A . Suppose A meets λ cosets of H. |A + A| ≥ ( 2 λ − 1 ) | H | Two extreme cases: WIN! Generic case: H = { 0 } −→ | 2 A| ≥ 2 |A| − 1 , No increase in size! Large Subgroup Problem A subgroup −→ | 2 A| = |A| ←−

Background Basic Problem The P 2 -path Some examples References ◮ 2 A = A + A is a union of H -cosets. ◮ 3 A is a union of H -cosets. ◮ G ∗ = G / H , A ∗ = A / H , it is enough to show that 3 A ∗ = G ∗ . ◮ 2 A ∗ has a trivial stabilizer in G ∗ . ◮ Set Y = | G / H | = | G ∗ | and λ = |A / G | . ◮ λ | H | ≥ |A| ≥ η | G | � λ ≥ ⌈ η Y ⌉ . ◮ If λ + ( 2 λ − 1 ) > | G ∗ | then 3 A ∗ = G ∗ . When 3 ⌈ η Y ⌉ ≥ Y + 2, we are done! 5 When Y ≥ 3 η − 1, we are done! If Y = 3 k or Y = 3 k + 1, we are done!

Background Basic Problem The P 2 -path Some examples References A first result Theorem Assume [ C 0 ] and [ C 1 ] . If |A / G | > 2 / 5 = 0 . 4 then 3 A = G. Difficulty with G ∗ = Z / 5 Z and A ∗ = { 0 , 1 } . We have 3 A ∗ � G ∗ . C onditions : Recall ◮ [ C 0 ] A generates G . ◮ [ C 1 ] A intersects every subgroup of G of index 2 .

Background Basic Problem The P 2 -path Some examples References An unusual condition! But [possibly] NOT C onditions : every coset!! ◮ [ C 2 ] A meets every subgroup of index 5 . ◮ [ C 3 ( Y 0 ) ] Given any subgroup H ⊂ G of index Y ≤ Y 0 , Given any coset u + H , A ∪ 2 A intersects u + H . Where does it come from? Again from sieve! From the weighted sieve in fact. It is a reformulation of Brun-Titchmarsh inequality for cosets but very different consequences!! Unclear for ray class group, work in progress –

Background Basic Problem The P 2 -path Some examples References A new (easy!) lemma Lemma Let G ∗ finite abelian. Assume A ∗ ∪ 2 A ∗ = G ∗ and 3 A ∗ � G ∗ . Then 1. 0 � A ∗ and A ∗ = −A ∗ , 2. The stabilizer of A ∗ is 0, 3. A ∗ ∩ 2 A ∗ = ∅ , Sum-free set! 4. |A ∗ | ≤ ( | G ∗ | + 1 ) / 3 , 5. 3 A ∗ = G \ { 0 } . The bound |A ∗ | ≤ ( | G ∗ | + 1 ) / 3 is optimal in general. But in specific cases? Large sum-free sets −→ (Yap, 1975), (Green & Ruzsa, 2005).

Background Basic Problem The P 2 -path Some examples References Consequences Theorem Assume [ C 0 ] , [ C 1 ] , [ C 2 ] and [ C 3 ( 5 )] . If |A / G | > 3 / 8 = 0 . 375 then 3 A = G. Theorem Assume [ C 0 ] , [ C 1 ] , [ C 2 ] and [ C 3 ( 5 )] . Assume the 2-part of G is ≃ ( Z / 2 Z ) r . If |A / G | > 4 / 11 = 0 . 3636 . . . then 3 A = G. That’s [p | q = ⇒ p ≡ 3 [ 4 ] ] for ( Z / q Z ) × or ( Z / 4 q Z ) × .

Background Basic Problem The P 2 -path Some examples References Elements of proof ◮ Y = | G / H | = 8 is the difficulty. ◮ A ∗ = A / G has exactly three elements. ◮ G ∗ can be Z / 8 Z , Z / 2 Z × Z / 4 Z or ( Z / 2 Z ) 3 . ◮ Rule out ( Z / 2 Z ) 3 : A ∗ is a basis of G ∗ . ◮ Condition of 2-part: rules out other cases.

Background Basic Problem The P 2 -path Some examples References In general ◮ In G ∗ = Z / 8 Z , A ∗ = { 1 , 4 , 7 } and A ∗ = { 3 , 4 , 5 } have to be ruled out. ◮ In G ∗ = Z / 2 Z × Z / 4 Z , A ∗ = { ( 0 , 1 ) , ( 0 , 3 ) , ( 1 , 0 ) } , A ∗ = { ( 1 , 1 ) , ( 1 , 3 ) , ( 1 , 0 ) } , A ∗ = { ( 1 , 1 ) , ( 1 , 3 ) , ( 0 , 2 ) } , A ∗ = { ( 1 , 1 ) , ( 1 , 3 ) , ( 1 , 0 ) } . ◮ The second does not touch Z / 2 Z × { 0 , 2 } . ◮ The fourth does not touch { 0 } × Z / 4 Z .

Background Basic Problem The P 2 -path Some examples References Examples F or the upper bound : G ∗ = Z / ( 3 k + 2 ) Z , A ∗ = [ k + 1 , 2 k + 1 ] , Then 2 A ∗ = [ 2 k + 2 , 3 k + 1 ] ∪ [ 0 , k ] , |A ∗ | = k + 1 = ( | G ∗ | + 1 ) / 3. F or the lower bound : Numerical experiments when G is Z /ℓ Z . Here are the sizes of the possible sets A ∗ : ℓ 8 11 17 18 19 20 21 22 23 24 25 |A ∗ | 3 4 6 6 6 6,7 6 7 8 7,8 8 ℓ 26 27 28 29 |A ∗ | 7,8,9 8 8,9 8,10

Background Basic Problem The P 2 -path Some examples References F or the lower bound : Some examples: A ∗ = { 1 , 4 , 7 } mod 8, A ∗ = { 1 , 3 , 5 , 12 , 14 , 16 } mod 17, A ∗ = { 1 , 3 , 7 , 12 , 16 , 21 , 25 , 27 } mod 28, A ∗ = { 1 , 3 , 5 , 12 , 14 , 16 , 23 , 25 , 27 } mod 28. A large cardinality example is given by C ∪ ( − C ) where C = { 1 , 3 , 5 , 7 , 9 , 20 , 22 , 24 } mod 48 . A large cardinality example is given by C ∪ ( − C ) where C = { 1 , 3 , 5 , 7 , 9 , 11 , 17 } mod 49 .

Background Basic Problem The P 2 -path Some examples References Green, Ben, & Ruzsa, Imre Z. 2005. Sum-free sets in abelian groups. Israel J. Math., 147 , 157–188. Ramaré, O., & Srivastav, Priyamvad. 2019. Products of primes in arithmetic progressions. To appear in Int. Journal of Number Theory, 17 pp. Appendix by O. Serra. Ramaré, O., & Walker, Aled. 2018. Products of primes in arithmetic progressions: a footnote in parity breaking. J. Number Theory of Bordeaux, 30 (1), 219–225. Yap, H. P . 1975. Maximal sum-free sets in finite abelian groups. V. Bull. Austral. Math. Soc., 13 (3), 337–342. Zaman, Asif. 2016. On the least prime ideal and Siegel zeros. Int. J. Number Theory, 12 (8), 2201–2229. Zaman, Asif. 2017. Bounding the least prime ideal in the Chebotarev density theorem. Funct. Approx. Comment. Math., 57 (1), 115–142.