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The quantitative behaviour of polynomial orbits on nilmanifolds
Number Theory · Submitted 22 Sep 2007
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim More specifically we show that there is a factorization $g = \epsilon g'\gamma$, where $\epsilon(n)$ is "smooth", $\gamma(n)$ is periodic and "rational", and $(g'(a),g'(a+d),\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $\delta$) inside some subnilmanifold $G'/\Gamma'$ of $G/\Gamma$, for all sufficiently dense arithmetic progressions $a,a+d,\ldots,a+d(l-1)$ inside $\{1,..,N\}$.
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The abstract, sentence by sentence
A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\Gamma$. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = \epsilon g'\gamma$, where $\epsilon(n)$ is "smooth", $\gamma(n)$ is periodic and "rational", and $(g'(a),g'(a+d),\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $\delta$) inside some subnilmanifold $G'/\Gamma'$ of $G/\Gamma$, for all sufficiently dense arithmetic progressions $a,a+d,\ldots,a+d(l-1)$ inside $\{1,..,N\}$. Our bounds are uniform in $N$ and are polynomial in the error tolerance delta. In a subsequent paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from our earlier paper "Linear equations in primes".
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\Gamma$. | background 1.00 | 0.02 | 0.04 | asserted 1.00 |
| s1 | In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\ldots,g(N))$ in a nilmanifold. | contribution 1.00 | 0.95 | 0.39 | asserted 0.99 |
| s2 | More specifically we show that there is a factorization $g = \epsilon g'\gamma$, where $\epsilon(n)$ is "smooth", $\gamma(n)$ is periodic and "rational", and $(g'(a),g'(a+d),\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $\delta$) inside some subnilmanifold $G'/\Gamma'$ of $G/\Gamma$, for all sufficiently dense arithmetic progressions $a,a+d,\ldots,a+d(l-1)$ inside $\{1,..,N\}$. | result 0.85 | 0.96 | 0.81 | asserted 1.00 |
| s3 | Our bounds are uniform in $N$ and are polynomial in the error tolerance delta. | result 0.91 | 0.93 | 0.12 | asserted 1.00 |
| s4 | In a subsequent paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from our earlier paper "Linear equations in primes". | future work 1.00 | 0.10 | 0.04 | asserted 0.90 |
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Original abstract
A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\Gamma$. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = \epsilon g'\gamma$, where $\epsilon(n)$ is "smooth", $\gamma(n)$ is periodic and "rational", and $(g'(a),g'(a+d),\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $\delta$) inside some subnilmanifold $G'/\Gamma'$ of $G/\Gamma$, for all sufficiently dense arithmetic progressions $a,a+d,\ldots,a+d(l-1)$ inside $\{1,..,N\}$. Our bounds are uniform in $N$ and are polynomial in the error tolerance delta. In a subsequent paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from our earlier paper "Linear equations in primes".
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References
How the literature uses it
9 arXiv papers cite this in 91 sentences
Papers citing it · 53 sentences without a confident label
Builds on it 2 papers
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This more general result can be deduced from [ref] in a somewhat routine, though slightly tedious, manner.
An arithmetic regularity lemma, associated counting lemma, and applications · Proof of the counting lemma · 0.92
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The material in this section is a generalisation of [ref] , and particularly [ref] , to the context of polynomial maps $(H_I →G_I)$ (there matters were discussed only in the case $(H →G_I)$).
An inverse theorem for the Gowers U^{s+1}[N]-norm · Polynomial algebra · 1.00
Uses its method or tool 6 papers
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Instead, we adapt a quantitative equidistribution result that was recently obtained by Green and Tao ( [ref] ).
A Hardy field extension of Szemeredi's Theorem · Results in combinatorial language · 0.95
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This argument can be adapted to the non-Abelian setup we are interested in, the main reason being that “Weyl type" sums involving polynomial block sequences of fixed degree on nilmanifolds can be effectively estimated using a rather sophisticated application of the van der Corput difference trick (this is done in [ref] ), and with a bit of care one can piece together these estimates to get usable results.
Equidistribution of sparse sequences on nilmanifolds · A model equidistribution result · 0.82
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The main tool here is a dimension reduction argument and a factorisation of nilsequences similar to that appearing in [ref] .
An arithmetic regularity lemma, associated counting lemma, and applications · Proof of the arithmetic regularity lemma · 1.00
Compares with it 4 papers
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The proof is (unsurprisingly) extremely similar to that of [ref] .
An arithmetic regularity lemma, associated counting lemma, and applications · Properties of polynomial sequences · 0.87
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This space is precisely the same space as the one considered in [ref] .
An inverse theorem for the Gowers U^{s+1}[N]-norm · Nilcharacters and symbols in one and several variables · 0.84
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So unlike the more refined result stated in [ref] , for the result we state here there is no reason to impose restrictions on the Malcev basis we use (or even refer to it).]
A Hardy field extension of Szemeredi's Theorem · The case of a connected group · 1.00
Cites it as background 3 papers
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In this subsection we review the basic notation and concepts involved, so as to set this paper in a workable context, but we omit several details, for which we refer the reader to [ref] .
Convergence results for systems of linear forms on cyclic groups, and periodic nilsequences · Nilmanifolds and polynomial sequences · 0.90
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For all concepts and notation in connection with nilmanifolds and nilsequences that remain undefined in this section we refer to [ref] and its companion paper [ref] .
Correlations of the divisor function · Non-correlation of the $W$-tricked divisor function with nilsequences · 0.85
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In [ref] , much emphasis was placed on the notion of an equidistributed polynomial sequence $g : →G$: one for which the orbit $(g(n)Γ)_n ∈[N]$ is close to equidistributed on $G/Γ$.
An arithmetic regularity lemma, associated counting lemma, and applications · Properties of polynomial sequences · 0.89
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