A bilinear Bogolyubov theorem

Bienvenu, Pierre-Yves; Lê, Thái Hoàng

doi:10.48550/arxiv.1711.05349

Cited by 3 publications

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“…In the course of thinking about that, we proved a more combinatorial statement that can also be thought of as a bilinear analogue of Bogolyubov's method [5]. That version was discovered independently Bienvenu and Lê [2].…”

Section: Introductionmentioning

confidence: 77%

“…Roughly speaking, Green, Tao and Ziegler use the U 3 inverse theorem for each a such that ∂ a f U 3 is large, and then prove that the corresponding generalized quadratic phase functions "line up in a linear way". We use the U 2 inverse theorem (which is a very simple calculation using Fourier expansions) for each (a, b) such that ∂ a,b U 2 is large and then prove that the characters we obtain "line up in a bilinear way". We start with a rather weak bilinearity property and gradually strengthen it: this is the sense in which our proof requires a detailed study of approximate bilinearity.…”

Section: Overview Of the Proofmentioning

confidence: 99%

“…These can be used to prove a stability result: an approximate bihomomorphism on B agrees almost everywhere with an exact bihomomorphism. (7) An exact bihomomorphism defined on a high-rank bilinear Bohr set can be extended to the whole of G 2 . Moreover an exact bihomomorphism on G 2 is given by a formula of the form x.Ay + T y + F(x), where A is a linear map from F n p to F n p , T is a linear map from F n p to F p , and F is an arbitrary function from F n p to F p .…”

Section: Overview Of the Proofmentioning

confidence: 99%

“…The main difference is that in the linear case there are tools that allow one to pass in one step from a function that "respects many additive quadruples" in an arbitrary dense set to a Freiman homomorphism on a dense subset of that set. We do not know how to prove the corresponding statement for bilinear functions without several steps (roughly steps (2) to (6) in the scheme above) in which one deals with approximate bihomomorphisms. Another difference is that, for reasons that we shall explain later, our bilinear version of Bogolyubov's method describes mixed single convolutions to within a small L 2 error, whereas the linear version describes double convolutions to within a small L ∞ error.…”

Section: Overview Of the Proofmentioning

confidence: 99%

“…For the final step, we apply Cauchy-Schwarz one more time, to the inner product that we have just obtained. The square of the first term is E P f (P) 2 2 , and the square of the second works out as E h(P 2 )−h(P 3 )=h(P 4 )−h(P 5 )=h(P 6 )−h(P 7 )=h(P 8 )−h(P 9 ) w(P 2 )=w(P 3 ),w(P 4 )=w(P 5 ),w(P 6 )=w(P 7 ),w(P 8 )=w(P 9 )…”

Section: 2mentioning

confidence: 99%

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A quantitative inverse theorem for the $U^4$ norm over finite fields

Gowers,

Milićević

2017

Preprint

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A remarkable result of Bergelson, Tao and Ziegler [1,18] implies that if c > 0, k is a positive integer, p ≥ k is a prime, n is sufficiently large, and f :where ω = exp(2πi/p) and c ′ > 0 is a constant that depends on c, k and p only. A version of this result for low-characteristic was also proved by Tao and Ziegler [19]. The proofs of these results do not yield a lower bound for c ′ . Here we give a different proof in the high-characteristic case when k = 4, which enables us to give an explicit estimate for c ′ . The bound we obtain is roughly doubly exponential in the other parameters.

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Section: Introductionmentioning

confidence: 77%

Section: Overview Of the Proofmentioning

confidence: 99%

Section: Overview Of the Proofmentioning

confidence: 99%

Section: Overview Of the Proofmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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A quantitative inverse theorem for the $U^4$ norm over finite fields

Gowers,

Milićević

2017

Preprint

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Linear and Quadratic Uniformity of the Möbius Function Over

Bienvenu

Lê

2019

Mathematika

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We examine correlations of the Möbius function over Fq [t] with linear or quadratic phases, that is, averages of the formfor an additive character χ over Fq and a polynomial Q ∈ Fq[x0, . . . , xn−1] of degree at most 2 in the coefficients x0, . . . , xn−1 of f = i 0 if Q is linear and O q −n c for some absolute constant c > 0 if Q is quadratic.The latter bound may be reduced to O(q −c ′ n ) for some c ′ > 0 when Q(f ) is a linear form in the coefficients of f 2 , that is, a Hankel quadratic form, whereas for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem.

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A bilinear Bogolyubov-Ruzsa lemma with poly-logarithmic bounds

Hosseini,

Lovett

2018

Preprint

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The Bogolyubov-Ruzsa lemma, in particular the quantitative bound obtained by Sanders, plays a central role in obtaining effective bounds for the U 3 inverse theorem for the Gowers norms. Recently, Gowers and Milićević applied a bilinear Bogolyubov-Ruzsa lemma as part of a proof of the U 4 inverse theorem with effective bounds. The goal of this note is to obtain a quantitative bound for the bilinear Bogolyubov-Ruzsa lemma which is similar to that obtained by Sanders for the Bogolyubov-Ruzsa lemma.We show that if a set A ⊂ F n × F n has density α, then after a constant number of horizontal and vertical sums, the set A contains a bilinear structure of codimension r = log O(1) α −1 . This improves the result of Gowers and Milićević, who obtained a similar statement with a weaker bound of r = exp(exp(log O(1) α −1 )), and by Bienvenu and Lê, who obtained r = exp(exp(exp(log O(1) α −1 ))).

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A bilinear Bogolyubov theorem

Cited by 3 publications

References 0 publications

A quantitative inverse theorem for the $U^4$ norm over finite fields

A quantitative inverse theorem for the $U^4$ norm over finite fields

Linear and Quadratic Uniformity of the Möbius Function Over

A bilinear Bogolyubov-Ruzsa lemma with poly-logarithmic bounds

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