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The norm of polynomial maps

This excerpt from https://arxiv.org/abs/1311.6170 summarizes basic facts on the norm of polynomial maps.

Recall that the norm $\Vert \theta \Vert _{R/Z}\in [0,1/2]$ of a real number $\theta$ is the distance from $\theta$ to the closest integer.

Let $g\colon \mathbb Z^t\to \mathbb R$ be a polynomial map. It can be written in the form $g(\mathbf n)= \sum _{\mathbf j} g_{\mathbf j} \left( \begin{array}{c}\mathbf n \\ \mathbf j\end{array} \right)$ where $\left( \begin{array}{c}\mathbf n \\ \mathbf j\end{array} \right)$ is the product of binomial coefficients $\left( \begin{array}{c}n_1 \\ j_1\end{array} \right)\cdot \dots \cdot\left( \begin{array}{c}n_t \\ j_t\end{array} \right)$. Given an integer $N\ge 1$, the $C^\infty [N]^t$-norm of $g$ is defined as: $\Vert g \Vert _{C^\infty [N]^t } := \max _{\mathbf j } \Bigl\{ N^{|\mathbf j|} \Vert g_{\mathbf j}\Vert _{R/Z} \Bigr\} .$ One could more generally consider integers $N_1,\dots ,N_t$ and the associated norm $C^\infty ([N_1]\times\dots\times [N_t] )$ but in practice we only consider the case $N_1=\dots =N_t$ at least up to bounded factors. (This norm and the one below are defined on p.4.)

Of course one could consider the expansion $g(\mathbf n)= \sum _{\mathbf j} g_{\mathbf j}^* \mathbf n ^{\mathbf j }$ and consider a variant:$\Vert g \Vert _{C_*^\infty [N]^t } := \max _{\mathbf j } \Bigl\{ N^{|\mathbf j|} \Vert g^*_{\mathbf j}\Vert _{R/Z} \Bigr\} .$ These two norms are essentially the same in the following sense (note that we shall often modify a polynomial by its $O(1)$-multiple so the modification $g\leadsto Qg$ is nothing):

Lemma 2.1. Let $g$ be of total degree $d$. Then there is some $Q=O_{d,t}(1)$ such that $\Vert Q g \Vert _{C^\infty _* [N]^t} \ll _{d,t}\Vert g \Vert _{C^\infty [N]^t}$ and vice versa.

For the proof, observe that the polynomial$\left( \begin{array}{c}\mathbf n \\ \mathbf j \end{array} \right)$ is a $\mathbb Q$-linear combination of $\mathbf n ^{\mathbf k}$'s with degrees $|\mathbf k|\le |\mathbf j|$ and vice versa. This shows that $g^*_{\mathbf j}$ is a $\mathbb Q$-linear combination of $g_{\mathbf k}$'s with $|\mathbf k|\ge |\mathbf j|$ (note the inequality!) and vice versa. Let $Q$ be a common multiple of the coefficients appearing there, which we can take depending only on $t$ and $d$. Then $Qg^*_{\mathbf j}$ is a $\mathbb Z$-linear combination of $g_{\mathbf k}$'s of degree equal or higher with coefficients bounded depending only on $d,t$. Suppose $M=\Vert g \Vert _{C^\infty [N]^t}$. This means $\Vert g_{\mathbf k}\Vert _{R/Z} \le \frac M{N^{|\mathbf k| } }$. We deduce $\Vert Qg^*_{\mathbf j}Vert _{R/Z} = O( \frac M{N^{|\mathbf j| } } + \frac{M}{N^{|\mathbf j| +1}}+\cdots )$ This completes the proof. ◼️

Proposition 2.3. (polynomial Vinogradov) Let $g\colon \mathbb Z^t\to \mathbb R$ be a polynomial map such that $\Vert g(\mathbf n)\Vert _{R/Z} \le \varepsilon$ for at least $\delta N^t$ values of $\mathbf n\in [N]^t$, where $\varepsilon < \delta / 10$. Then there is some $Q\ll \delta ^{-O(1)}$ such that $\Vert Qg \Vert _{C^\infty [N]^t} \ll \delta ^{-O(1)}\varepsilon$, i.e., we have $\Vert Qg_{\mathbf j}\Vert _{R/Z} \ll \frac{\delta ^{-O(1)} \varepsilon}{N^{|\mathbf j|} }$ for all indices $\mathbf j$.

Note that by the previous lemma a similar result holds for $g^*_{\mathbf j}$ as well. This proposition roughly says "if the value of a polynomial $g$ is often close to integers, then the (especially higher-order) coefficients of $g$ are extremely close to integers."

Proposition 2.3 is a vitally important tool for us and so one should memorize it well!

The next assertion says that if you know the behavior of a polynomial map after dilation and translation, you know the polynomial map itself quite well.

Proposition 8.4 (in Quantitative behaviour of polynomial sequences). Let $a_i,b_i\in \mathbb Q$ with $b_i\neq 0$ be such that their deniminators are bounded by $Q$ and the numerators of $b_i$ are also bounded by $Q$. Assume the numerators of $a_i$ are bounded by $QN$.

Let $g\colon \mathbb Z^t \to \mathbb R$ be a polynomial map and consider the polynomial map $\tilde g$ defined by $\mathbf n=(n_1,\dots ,n_t)\mapsto g(a_1+b_1n_1,\dots ,a_t+b_tn_t)$. Then there is an integer $Q'\ll _{d,t} Q^{O_{d,t}(1)}$ such that $\Vert Q'\cdot g\Vert _{C^\infty [N]^t} \ll _{d,t} Q^{O_{d,t}(1) } \Vert \tilde g\Vert _{C^\infty [N]^t} .$

In loc. cit. they also assume the numerators of $a_i$ are bounded by $Q$, as opposed to $QN$ in our formulation. So let us check the proof loc. cit. actually shows the current assertion.

If we set $M:=\Vert g\Vert _{C_*^\infty [N]^t}$, we can write $\tilde g (\mathbf m) = \sum _{\mathbf j} \tilde g^*_{\mathbf j} \mathbf m ^{\mathbf j }$ with $\Vert g^*_{\mathbf j }\Vert _{R/Z} \le \frac{M}{N^{|\mathbf j|} }$. Since $g(\mathbf n)= \tilde g (\frac{n_1-a_1}{b_1},\cdots )$ we can write $g(n_1,\dots ,n_t)= \sum _{\mathbf j} \tilde g^*_{\mathbf j} \left( \frac{n_1-a_1}{b_1}\right) ^{j_1} \cdot \dots \cdot \left( \frac{n_t-a_t}{b_t}\right) ^{j_t} .$ It follows that $g^*_{\mathbf j }$ is a sum of terms of the form $\tilde g^* _{\mathbf k} \frac{\mathbf a ^{\mathbf k - \mathbf j } }{\mathbf b ^{\mathbf k} }$ with $\mathbf k \ge \mathbf j$ entrywise and the number of terms being bounded in terms of $d,t$.

Clear the denominators appearing in such terms by multiplying an integer $\le Q^{O_{d,t}(1) }$. Then we are left with the following, where $\mathrm{num}(-)$ is a temporary notation to denote the numerator of a rational number: $\Vert \tilde g^*_{\mathbf k} \mathrm{num}(\mathbf a ^{\mathbf k - \mathbf j } ) \mathrm{num}(\mathbf b^{\mathbf k} ) ) \Vert _{R/Z } \le \Vert\tilde g^*_{\mathbf k}\Vert _{R/Z} \cdot Q^k N ^{k-j} \le \frac{M}{N^k} Q^kN^{k-j}$ which gives a desired bound for $g^*_{\mathbf j}$. ◼️

Lastly let us record the quantitative Leibman theorem.

Theorem 8.6. (proof completed in the Erratum) Let $G/\Gamma$ be a nilmanifold of dimension $m$ with a $\frac 1\delta$-rational Malcev basis. If a polynomial map $g\colon \mathbb Z^t \to G$ is not $\delta$-equidistributed on $[N]^t$, then there is a non-trivial horizontal character $\psi$ with $\Vert \psi\Vert \ll \delta ^{-O_{d,m,t}(1) }$ such that $\Vert\psi \circ g \Vert _{C^\infty [N]^t} \ll \delta ^{-O_{d,m,t}(1)}.$

Note that the converse statement (existence of such a character $\Rightarrow$ non-equidistribution) is somewhat straightforward to prove. One can even draw a picture!