Quantitative Leibman Theorem --- refinement by Tao-Teäväinen
The Quantitative Leibman Theorem, due to Green-Tao, is the follwoing statement:
Quantitative Leibman Theorem.
$G/\Gamma$ an m-dimensional nilmanifold of degree $\le d$ equipped with a $1/\delta$-rational Malcev basis. $g\colon \mathbb Z \to G$ a polynomial map. If it is not $\delta$-equidistributed on $[N]$, then there is a non-zero horizontal character $\psi \colon G_{ab} \to \mathbb R $ with $ |\psi |\ll \delta ^{-O_{m,d}(1)}$ such that
\[ \Vert \psi \circ g \Vert _{C^\infty [N]} \ll \delta ^{-O_{m,d}(1)}. \]
(Theorem 2.9 (p.480) of [GT] := "Quantitative behaviour of polynomial orbits on nilmanifolds"https://annals.math.princeton.edu/2012/175-2/p02
)
Green and Tao didn't need any explicit control of the constants $O_{m,d}(1)$.
https://arxiv.org/abs/2107.02158
In the above preprint [TT] of Tao-Teravainen, they use Manners' version of Inverse Gowers Theorem. It gives a wonderfully stronger output if one feeds it with a slightly stronger input than Green-Tao's former work. To guarantee this stronger input, they need a nice enough explicit control of $O_{m,d}(1)$. By reviewing the proof of Quantitative Leibman Theorem, they establish a refined statement:
Quantitative Leibman Theorem is true with $O_{m,d}(1)=\exp ( (2m )^{O_d(1)} )$.
(Theorem A.3 of [TT])
Green-Tao [GT] used Quantitative Leibman Theorem to deduce Factorization Theorem, which is also an indispensable tool when dealing with polynomial sequences. Its statement is too complicated for a blog post---let me just mention that it decomposes a given polynomial sequence $g$ as the pointwise product $g=\varepsilon g' \gamma $ of smooth, equidistributed and periodic sequences.
In the statement there is a quantity M satisfying the following estimate in terms of initial data $M_0>0$ and $A>0$:
$M_0 < M < M_0 ^{O_{A,m,d}(1)}$.
(See Theorem 1.19 of [GT])
Again Tao-Teravainen needed a nice enough bound of the constant $O_{A,m,d}(1)$. By refining the deduction by Green-Tao, they established the following.
One can take $O_{A,m,d}(1)= A^{(2+m)^{O_d(1)}} $.
(Theorem A.6 of [TT])
To summarize,
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Quantitative Leibman Theorem (refined in [TT] by reviewing the proof in [GT])
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|| [GT]'s deduction with some extra care ([TT])
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V
Factorization Theorem (refined)
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Multiparameter statements
Actually, there are multi-parameter versions of Quantitative Leibman Theorem and Factorization Theorem. The initial proof in [GT] had some flaw but it was fixed in the Erratum =: [Err].
https://arxiv.org/abs/1311.6170v3
Quantitative Leibman Theorem (multiparameter, "equal parameter case").
$g$ is now a polynomial map $\mathbb Z^t \to G$. If it is not $\delta $-equidistributed on $[N]^t$, then the same conclusion holds with exponents $O_{d,m,t}(1)$.
(See §3 of [Err])
This equal parameter case can be deduced from the one-parameter Quantitative Leibman Theorem with relatively little effort.
Once we have this, [GT] already shows that Factorization follows.
Factorization Theorem stays true for $g\colon \mathbb Z^t \to G$ with the estimate $M_0<M<M_0^{O_{A,m,d}(1) }$.
(See Theorem 10.2 of [GT]. (Is it really true that the exponent doesn't depend on $t$?) )
To summarize,
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Quantitative Leibman Theorem
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|| relatively little effort [Err]
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V
Multi-parameter Quantitative Leibman (equal parameter case)
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|| [GT]
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V
Multi-parameter Factorization Theorem (equal parameter)
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Refinement of the multiparameter statements?
One imagines that the following scheme works:
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Quantitative Leibman (refined in [TT])
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|| [Err]'s method
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V
Multi-parameter Quantitative Leibman (refined, equal parameters) (?)
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|| [GT]'s method with care (= [TT])
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V
Multi-parameter Factorization Theorem (refined, equal parameters)
(?)
("(?)" means that it has not been established in the literature.)
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I kind of checked that this scheme works. I'll write some detail when I get ready.
