Notation

We will use the following notation for the disk and the circle. ${\mathbb D}(a,r) = \{ z \in {\mathbb C} \colon |z-a| < r \}$ , ${\mathbb T}(a,r) = \partial {\mathbb D}(a,r)$ . We also write ${\mathbb D} = {\mathbb D}(0,1)$ , ${\mathbb T} = \partial {\mathbb D}$ for the unit disk and unit circle.

We write $\mathrm {d}|z|$ to denote integration with respect to arc-length measure, and $\mathrm {d} A(z)$ denotes integration with respect to area measure (both are not normalized). That is, if ${\Gamma }$ is a (rectifiable) curve, and G is a domain, then

$$\begin{align*}\int_{\Gamma} 1 \, \mathrm{d} |z| = \textrm{length}({\Gamma}), \qquad \int_G 1 \, \mathrm{d} A(z) = \textrm{area}(G). \end{align*}$$

3.1 The $\boldsymbol{\ell}^{\boldsymbol{\infty}} $ constraint

Let $t> 0$ . We first consider the (easier) problem of the asymptotic volume of the set

$$\begin{align*}\left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \sup_{n\in{\mathbb N}} |P_{\textbf{c}}(n) A^{-n}| \le t \right\}. \end{align*}$$

Using theorems by Ball and Vaaler, and some approximations (see Section 4 for the details), we will arrive at the following result

$$\begin{align*}\frac{t^{d+1}}{\sqrt{\det {\mathcal M}}} \le \operatorname{\mathrm{Vol}}\left( \Big\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \sup_{n\in {\mathbb N}} | P_{\mathbf{c}}(n) r^n | \le \tfrac12 t \Big\} \right) \le \frac{C_1^{d+1} t^{d+1}}{\sqrt{\det {\mathcal M}}}, \end{align*}$$

where $C_1> 0$ is an absolute constant, and

$$\begin{align*}{\mathcal M} = {\mathcal M}(d;r) = \left[ \sum_{n=0}^\infty \binom{n}{j} \binom{n}{k} r^{2n} \right]_{j,k=0}^{d}. \end{align*}$$

Observe that by the orthogonality of the monomials

$$\begin{align*}\begin{aligned} \sum_{n=0}^\infty P_1(n) P_2(n) r^{2n} & = \frac{1}{2 \pi r} \int_{r {\mathbb T}} \left[ \sum_{m=0}^\infty P_1(m) w^m \right] \left[ \sum_{n=0}^\infty P_2(n) \bar{w}^n \right] \mathrm{d} |w| \\ & = \frac{1}{2 \pi r} \int_{r {\mathbb T}} g_{P_1}(w) \overline{g_{P_2}(w)} \, \mathrm{d} |w|, \end{aligned} \end{align*}$$

where the integrals are with respect to arc-length.

By a change of variables, using the fractional linear transformation $w = z/(1-z)$ , we conclude that

$$\begin{align*}{\mathcal M} = \left[ \frac{1}{2 \pi r} \int_{{\mathbb T}_1} w^j \bar{w}^k \, \mathrm{d} |w| \right]_{j,k=0}^{d} \end{align*}$$

is a moment matrix, where

$$\begin{align*}{\mathbb D}_1 = {\mathbb D}\left(\frac{r^2}{1-r^2},\frac{r}{1-r^2}\right), \quad {\mathbb T}_1 = \partial {\mathbb D}_1 = {\mathbb T}\left(\frac{r^2}{1-r^2},\frac{r}{1-r^2}\right). \end{align*}$$

It is known that (see, e.g., Appendix A)

$$\begin{align*}\det {\mathcal M} = \prod_{k=0}^d \| q_k \|^2_{L^2(\mu)}, \end{align*}$$

where $q_k$ is the monic orthogonal polynomial (of degree k) with respect to $\mu $ – the arc-length measure on ${\mathbb T}_1$ (weighted by $\frac {1}{2 \pi r}$ ). These polynomials and their norms are given explicitly by

$$\begin{align*}q_k(w) = \big( w - \tfrac{r^2}{1-r^2} \big)^k, \qquad \| q_k \|^2_{L^2(\mu)} = \frac1r \Big( \frac{r}{1-r^2} \Big)^{2k+1}, \end{align*}$$

hence

$$\begin{align*}\det {\mathcal M} = \frac{1}{r^{d+1}} \Big( \frac{r}{1-r^2} \Big)^{(d+1)^2}. \end{align*}$$

The critical threshold is achieved when $\frac {r}{1-r^2} = 1$ , that is $r = \varphi ^{-1}$ . Theorem 1.1 which concerns lattice points is obtained using a related theorem of Vaaler.

3.2 The $\boldsymbol{\ell}^{\boldsymbol{2}}$ constraint

Here we have, for $r < 1$ ,

$$\begin{align*}\frac{1}{2 \pi r} \int_{r {\mathbb T}} |g_P(z)|^2 \mathrm{d} |z| = \sum_{n=0}^\infty |P(n)|^2 r^{2n} = \| a_n \|^2_{\ell^2}, \end{align*}$$

and

$$\begin{align*}\frac{1}{\pi r^2} \int_{r {\mathbb D}} |g_P(z)|^2 \mathrm{d} A(z) = \sum_{n=0}^\infty \frac{|P(n)|^2}{n+1} r^{2n} = \Big\| \frac{a_n}{\sqrt{n+1}} \Big\|^2_{\ell^2}, \end{align*}$$

where we write $a_n = a_n(r) = P(n) r^n$ , and the second integral is with respect to planar Lebesgue measure.

Again making the change of variables $w = z/(1-z)$ , we find that

$$\begin{align*}\| a_n \|^2_{\ell^2} = \frac{1}{2 \pi r} \int_{{\mathbb T}_1} |h_{\textbf{c}}(w)|^2 \mathrm{d} |w|, \qquad \Big\| \frac{a_n}{\sqrt{n+1}} \Big\|^2_{\ell^2} = \frac{1}{\pi r^2} \int_{{\mathbb D}_1} |h_{\textbf{c}}(w)|^2 \frac{\mathrm{d} A(w)}{|1+w|^2}, \end{align*}$$

where ${\mathbb D}_1$ and ${\mathbb T}_1$ are as above, and

$$\begin{align*}h_{\mathbf{c}}(w) = \sum_{k=0}^d c_k w^k. \end{align*}$$

Let $q_k$ be the (Szegő) monic orthogonal polynomial with respect to the measure $\mu $ as above, and $t_k$ be the (Bergman) monic orthogonal polynomial, with respect to weighted area measure

$$\begin{align*}\mathrm{d} \nu(w) = \frac{\mathrm{d} A(w)}{\pi r^2 |1+w|^2}\quad\text{on}\quad {\mathbb D}_1. \end{align*}$$

We are not aware of any simple formula for the $t_k$ s (see, e.g., [Reference Gustafsson, Putinar, Saff and Stylianopoulos10, Remark 8.1]). However, by results of Korovkin (see Section 5.1) we have, with $r = \varphi ^{-1}$ , and some $q \in (0,1)$ ,

$$\begin{align*}\| t_k \|^2_{L^2(\nu)} = \frac{1}{k+1} + O(q^k). \end{align*}$$

We obtain the correspondences

$$\begin{align*}\sum_{n=0}^\infty |P_{\mathbf{c}}(n)|^2 \varphi^{-2n} \le t^2 \iff \sum_{k=0}^d |\alpha_k(\mathbf{c})|^2 \| q_k \|^2_{L^2(\mu)} = \varphi \sum_{k=0}^d |\alpha_k(\mathbf{c})|^2 \le t^2, \end{align*}$$

where $h_{\mathbf {c}} = \sum _{k=0}^d \alpha _k(\mathbf {c}) q_k$ , and

$$\begin{align*}\sum_{n=0}^\infty \frac{|P(n)|^2}{n+1} \varphi^{-2n} \le t^2 \iff \sum_{k=0}^d |\beta_k(\mathbf{c})|^2 \| t_k \|^2_{L^2(\nu)} \le t^2, \end{align*}$$

where now $h_{\mathbf {c}} = \sum _{k=0}^d \beta _k(\mathbf {c}) t_k$ . The proof of the first part of Theorem 1.2 follows by observing that since $q_k$ are monic polynomials, if $h_{\mathbf {c}}$ has integer coefficients, then the first, in decreasing order of k, nonzero coefficient $\alpha _k(\mathbf {c})$ is the same integer. The second part of the theorem is deduced from a classical theorem of van der Corput on the existence of lattice points inside symmetric convex bodies with a large volume.

4.1 Generating functions

To an algebraic polynomial P we associate the generating function

(4.1) $$ \begin{align} g(z) = g_P(z) = \sum_{n=0}^\infty P(n) z^n, \end{align} $$

which converges for $|z| < 1$ . We also denote by $g_k$ the generating function of the binomial polynomial $\binom {\cdot }{k}$ . We leave the proof of the following claim as an exercise to the reader.

Claim 4.1. For $|z| < 1$ and $k\ge 0$ , we have

$$\begin{align*}g_k(z) = \frac{1}{1-z} \left( \frac{z}{1-z} \right)^k. \end{align*}$$

The above claim provides an analytic continuation of $g_P$ to a meromorphic function.

Claim 4.2. For any polynomial P, the function $g_P$ is analytic in ${\mathbb C} \setminus \{1\}$ with $g_P(\infty ) = 0$ , and in addition

(4.2) $$ \begin{align} g_P(z) = -\sum_{n=1}^\infty \frac{P(-n)}{z^n} \quad \text{for all } |z|> 1. \end{align} $$

Proof. Any polynomial P of degree d can be written in a unique way as

$$\begin{align*}P(x) = \sum_{k=0}^d c_k \binom{x}{k}, \end{align*}$$

hence it is sufficient to prove the claim for binomials. By Claim 4.1, for $k \in {\mathbb N}$ we have that

$$\begin{align*}g_k(z) = \sum_{n\in{\mathbb N}} \binom{n}{k} z^n = \frac{z^k}{(1-z)^{k+1}}, \qquad |z| < 1, \end{align*}$$

where the right hand side is analytic in $z \in {\mathbb C} \setminus \{1\}$ and vanishes at infinity. Let $\varepsilon \in (0,1)$ . Using the vanishing of $g_k$ at infinity, and Cauchy’s integral formula, we have

$$\begin{align*}\binom{n}{k} = \frac{1}{2\pi i} \int_{|w|=\varepsilon} \frac{w^k}{(1-w)^{k+1}} \frac{\mathrm{d} w}{w^{n+1}} = -\frac{1}{2\pi i} \int_{|w-1|=\varepsilon} \frac{w^k}{(1-w)^{k+1}} \frac{\mathrm{d} w}{w^{n+1}}. \end{align*}$$

We will use $w^z = \exp (z \log w)$ , where the principal branch of $\log $ , cut along the ray $(-\infty , 0]$ , is chosen. Observe that with this choice,

$$\begin{align*}Q(z) := -\frac{1}{2\pi i} \int_{|w-1|=\varepsilon} \frac{w^k}{(1-w)^{k+1}} \frac{\mathrm{d} w}{w^{z+1}}, \end{align*}$$

is an entire function with at most polynomial growth at infinity (this can be seen by choosing $\varepsilon $ of order $|z|^{-1}$ ). Hence by Liouville’s theorem Q is a polynomial. Since the polynomials Q and $\binom {\cdot }{k}$ coincide on the (infinite) set ${\mathbb N}$ , we conclude they are equal. In particular, for $n \in {\mathbb N}$ ,

$$\begin{align*}\binom{-n}{k} = -\frac{1}{2\pi i} \int_{|w-1|=\varepsilon} \frac{w^{k+n}}{(1-w)^{k+1}} \frac{\mathrm{d} w}{w}. \end{align*}$$

By making a change of variables $\zeta = 1/w$ , so that $\mathrm {d} w / w = -\mathrm {d} \zeta / \zeta $ , and shifting the contour we find

$$\begin{align*}\binom{-n}{k} = \frac{1}{2\pi i} \int_{|\zeta^{-1}-1|=\varepsilon} \frac{1}{(\zeta-1)^{k+1}} \frac{\mathrm{d} \zeta}{\zeta^n} = \frac{1}{2\pi i} \int_{|\zeta|=\varepsilon} \left[-\frac{1}{(\zeta-1)^{k+1}} \right] \frac{\mathrm{d} \zeta}{\zeta^n}. \end{align*}$$

Therefore, again using Cauchy’s formula, for sufficiently small $|\zeta |$ ,

$$\begin{align*}\sum_{n\ge 1} \binom{-n}{k} \zeta^{n-1} = -\frac{1}{(\zeta - 1)^{k+1}} \implies \sum_{n\ge 1} \binom{-n}{k} \zeta^{n} = -\frac{\zeta}{(\zeta - 1)^{k+1}}. \end{align*}$$

Finally, putting $z = 1/\zeta $ we obtain

$$\begin{align*}\sum_{n\ge 1} \binom{-n}{k} z^{-n} = -\frac{z^k}{(1-z)^{k+1}} = -g_k(z), \quad \text{ where } |z|> 1, \end{align*}$$

as claimed.

The following properties of the generating function are crucial for us.

Corollary 4.3. For a polynomial $P(z) = \sum _{k=0}^d c_k \binom {z}{k}$ of degree d, the function $g_P$ can be analytically continued to ${\mathbb C} \setminus \{1\}$ so that

$$\begin{align*}g_P(z) = \frac{1}{1-z} \sum_{n=0}^d c_k \left( \frac{z}{1-z} \right)^k, \quad z \in {\mathbb C} \setminus \{1\}. \end{align*}$$

Moreover, for $r> 0$ ,

$$\begin{align*}\frac{1}{2 \pi r} \int_{r {\mathbb T}} |g_P(w)|^2 \mathrm{d} |w| = \begin{cases} \sum_{n=0}^\infty |P(n)|^2 r^{2n}, & r < 1;\\ \sum_{n=1}^\infty |P(-n)|^2 r^{-2n}, & r> 1, \end{cases} \end{align*}$$

and

$$ \begin{align*} \frac{1}{\pi r^2} \int_{r {\mathbb D}} |g_P(w)|^2 \mathrm{d} A(w) & = \sum_{n=0}^\infty \frac{|P(n)|^2}{n+1} r^{2n},\quad \text{when } r < 1,\\ \frac{r^2}{\pi} \int_{(r {\mathbb D})^c} |g_P(w)|^2 \, \frac{\mathrm{d} A(w)}{|w|^4} & = \sum_{n=1}^\infty \frac{|P(-n)|^2}{n+1} r^{-2n},\quad \text{when } r> 1. \end{align*} $$

Proof. The first part follows from the previous claims. The second part follows by expanding $|g_P(w)|^2$ as in (4.1) and (4.2) in the variables $w, \overline {w}$ and observing that

$$\begin{align*}\frac{1}{2 \pi r} \int_{r {\mathbb T}} w^j \overline{w}^k \mathrm{d} |w| = r^{2 j} \delta_{j,k}, \quad \text{when } j,k \in {\mathbb Z}. \end{align*}$$

Similarly, the formulas with respect to area measure follow from

$$\begin{align*}\frac{1}{\pi} \int_{r {\mathbb D}} w^j \overline{w}^k \mathrm{d} A(w) = \frac{r^{2(j+1)}}{j+1} \delta_{j,k}, \quad \text{when } r<1 \text{ and } j,k \in {\mathbb N}, \end{align*}$$

and

$$\begin{align*}\frac{1}{\pi} \int_{(r {\mathbb D})^c} \frac{1}{w^{j}} \frac{1}{\overline{w}^{k}} \, \frac{\mathrm{d} A(w)}{|w|^4} = \frac{r^{-2(j+1)}}{j+1} \delta_{j,k}, \quad \text{when } r>1 \text{ and } j,k \in {\mathbb N}.\\[-40pt] \end{align*}$$

The next result follows from the above by polarization. We leave the details to the reader.

Corollary 4.4. For $j, k\ge 0$ we have,

$$\begin{align*}\sum_{n=0}^\infty \binom{n}{j} \binom{n}{k} r^{2n} = \frac{1}{2\pi r} \int_{r {\mathbb T}} g_j(w) \overline{g_k(w)} \mathrm{d} |w|, \quad r \in (0,1), \end{align*}$$

and

$$\begin{align*}\sum_{n=1}^\infty \binom{-n}{j} \binom{-n}{k} r^{-2n} = \frac{1}{2\pi r} \int_{r {\mathbb T}} g_j(w) \overline{g_k(w)} \mathrm{d} |w|, \qquad r> 1. \end{align*}$$

4.2 Tools for the $\boldsymbol{\ell}^{\boldsymbol{\infty}} $ case

Let $N, k\in {\mathbb N}$ with $N>k$ , and let $\Sigma $ be an $N \times k$ matrix of rank k. We use the following version of a result by Vaaler [Reference Vaaler15, Thms. 1 and 2], see also Gluskin-Kashin [Reference Gluskin and Kashin8].

Theorem 4.5. We have that

$$\begin{align*}V_N := \operatorname{\mathrm{Vol}}\left( \Big\{ \textbf{c} = (c_1, \dots, c_k) \in {\mathbb R}^{k} \colon \sup_{n\le N} \Big|\sum_{j=1}^k c_j \Sigma_{n,j} \Big| \le \tfrac12 \Big\} \right) \ge \frac{1}{\sqrt{\det \Sigma^t \Sigma}}. \end{align*}$$

Moreover, if $R \in {\mathbb N}$ is such that

$$\begin{align*}R^2 \cdot \det \Sigma^t \Sigma \le 1, \end{align*}$$

then there are at least $2R$ nontrivial solutions $\textbf {c} \in {\mathbb Z}^k$ to the system of inequalities

$$\begin{align*}\left| \sum_{j=1}^k c_j \Sigma_{n,j} \right| \le 1 \qquad \text{for all} \quad n\in \{1, \dots, N\}. \end{align*}$$

In the other direction we use a result of Ball [Reference Ball2] (also Gluskin-Kashin [Reference Gluskin and Kashin8]).

Theorem 4.6. We have that

$$\begin{align*}\operatorname{\mathrm{Vol}}\left( \Big\{ \textbf{c} = (c_1, \dots, c_k) \in {\mathbb R}^{k} \colon \sup_{n\le N} \Big|\sum_{j=1}^k c_j \Sigma_{n,j}\Big| \le 1 \Big\} \right) \le \frac{\sqrt{2}^{N - k}}{\sqrt{\det \Sigma^t \Sigma}}. \end{align*}$$

For the convenience of the reader we prove the following standard result in Appendix A.

Theorem 4.7. Let $\mu $ be a measure on ${\mathbb C}$ with infinite support and finite moments of all orders. Define the n-th moment matrix by

$$\begin{align*}{\mathcal M}(n;\mu) = \left[ M_{k,m} \right]_{k,m=0}^n, \qquad M_{k,m} = \int_{\mathbb C} z^j \bar{z}^k \, \mathrm{d} \mu(z). \end{align*}$$

Then

$$\begin{align*}\det {\mathcal M}(n;\mu) = \prod_{j=0}^n \left\| Q_j \right\|^2_{L^2(\mu)}, \end{align*}$$

where $Q_j$ are the orthogonal monic polynomials, with respect to $\mu $ .

Let $\eta \in (0,1)$ . Suppose ${\Gamma } = {\mathbb T}_1 \cup {\mathbb T}_2$ is the union of two circles whose radii are bounded between $\eta $ and $\frac {1}{\eta }$ , and their interiors are separated by at least $\eta $ and at most $\frac {1}{\eta }$ . Denote the $n \times n$ moment matrix, with respect to arc-length measure on ${\Gamma }$ , by ${\mathcal M}_{{\Gamma }}(n)$ . We will need an effective lower bound for the smallest eigenvalue of ${\mathcal M}_{{\Gamma }}(n)$ , which is given by

$$\begin{align*}\lambda_{n}^{-1} = \max \left\{ \frac{1}{2\pi} \int_{-\pi}^\pi \left| P(e^{i\theta}) \right|{}^2 \mathrm{d} \theta \, \colon \, \deg P = n, \,\, \int_{\Gamma} |P(z)|^2 |\mathrm{d} z| = 1 \right\}. \end{align*}$$

The following result is obtained by combining [Reference Widom16, Theorem 7.3 and Eq. (10.5)].

Proposition 4.8. Using the same notation as above, and under the same assumptions, we have

$$\begin{align*}\lambda_{n} \ge \exp(-C_{{\Gamma}} n), \end{align*}$$

where the constant $C_{{\Gamma }}> 0$ depends continuously on the Green function of the exterior of ${\Gamma }$ (and its derivatives).

4.2.1 Elementary results

We use the standard bound (that follows from the inequality $n! \ge (n/e)^n$ )

(4.3) $$ \begin{align} \binom{n}{k} \le \left( \frac{e n}{k} \right)^k, \qquad n> k \ge 1. \end{align} $$

Claim 4.9. Let $a> 0$ and $C \ge e$ , then we have

$$\begin{align*}\sup_{1\le x \le a/C} \left( \frac{a}{x} \right)^x \le C^{a/C}. \end{align*}$$

Proof. The function $x \log \tfrac {a}{x}$ is increasing on $[1,a/C]$ .

Lemma 4.10. For every $A> 1$ there exists a constant $C_A> 1$ such that if P is a polynomial of degree d satisfying $|P(n)| \le A^n$ for all $0 \le n \le C_A d$ , then $|P(n)| \le A^n$ for all $n \in {\mathbb N}$ .

Proof. The Lagrange interpolation formula states that

$$\begin{align*}P(x) = \sum_{j=0}^d P(j) L_j(x),\qquad \text{where} \quad L_j(x) = \prod_{\overset{0\le m\le d}{m\ne j}} \frac{x-m}{j-m}. \end{align*}$$

Notice that for $n \ge d+1$

$$\begin{align*}|L_j(n)| \le n^{d+1} \prod_{\overset{0\le m\le d}{m\ne j}} (j-m)^{-1} \le \frac{n^{d+1}}{[(\lfloor d/2\rfloor)!]^2} \le \left( \frac{6 n}{d} \right)^{d+1}. \end{align*}$$

Let $C_A$ be a constant depending only on A to be chosen later. Under the assumption of the lemma, we have that $|P(n)|\le A^n$ for all $0 \le n \le C_A d$ . In addition, for $n \ge d+1$

$$\begin{align*}|P(n)| \le \left( \frac{6 n}{d} \right)^{d+1} \sum_{j=0}^d A^j \le \frac{1}{A-1} \left( \frac{6 A n}{d} \right)^{d+1}. \end{align*}$$

If we choose $C_A$ large enough so that

$$\begin{align*}\mathrm{(i)}\,\, \exp(2/C_A) \le A \quad \mathrm{(ii)}\,\, 6 A C_A \le A^{C_A/2} \quad \mathrm{(iii)}\,\, (A-1)^{-1} \le A^{C_A/2}, \end{align*}$$

then we have that

$$\begin{align*}\frac{1}{A-1} \left( \frac{6 A n}{d} \right)^{d+1} \le A^n, \qquad \text{for all}\quad n \ge C_A d. \end{align*}$$

In particular, we may assume $C_A$ is decreasing with A.

A similar result holds for $P(-n)$ (e.g., define $Q(z) = B^{-1} P(-z -1)$ and apply the lemma). Hence, one can prove the following generalization. We leave the details to the reader.

Lemma 4.11. For every $A, B> 1$ there exists a constant $C_{A,B}> 1$ such that if P is a polynomial of degree d satisfying $|P(n)| \le A^n$ and $|P(-n)| \le B^n$ for all $0 \le n \le C_{A,B} \cdot d$ , then $|P(n)| \le A^n$ for all $n \in {\mathbb N}$ and $|P(-n)| \le B^n$ for all $n \in {\mathbb N}$ .

The following lemma will allow us to approximate the value of the determinant of the moment matrix.

Lemma 4.12. Let $A> 1$ , then for $L> C_A$ (where $C_A> e$ is the constant from Lemma 4.10), $n \in {\mathbb N}$ and $k,m\in \{0, \dots , n-1\}$ we have

$$\begin{align*}\sum_{j> L n} A^{-2j} \binom{j}{k} \binom{j}{m} \le \frac{A}{A-1} \exp((2 - L \log A) n). \end{align*}$$

Proof. We will prove the result for $k,m \ge 1$ , the other cases being similar. For $j> Ln$ , we have by (4.3) and Claim 4.9,

$$\begin{align*}A^{-2j} \binom{j}{k} \binom{j}{m} \le e^{k + m} A^{-2j} \frac{j^{k+m}}{k^k m^m} \le e^{2n} A^{-2j} L^{2j / L}. \end{align*}$$

Therefore, if $L> e$ is large enough so that $L^{1/L} < \sqrt {A}$ we get

$$\begin{align*}\sum_{j> L n} A^{-2j} \binom{j}{k} \binom{j}{m} \le e^{2n} \sum_{j > L n} A^{-j}, \end{align*}$$

and the result follows.

Similarly one can prove the following extension; we skip the details of the proof (one has to use $\left |\binom {-n}{k}\right | \le \binom {2n}{k}$ for $n\ge 0$ and $k \in \{0, \dots , n-1\}$ ).

Lemma 4.13. Let $A, B> 1$ , then for $L> C_{A,B}$ (where $C_{A,B}> e$ is the constant from Lemma 4.11), $n \in {\mathbb N}$ and $k,m\in \{0, \dots , n-1\}$ we have

$$\begin{align*}\begin{aligned} \sum_{j < -L n} & B^{2j} \binom{j}{k} \binom{j}{m} + \sum_{j> L n} A^{-2j} \binom{j}{k} \binom{j}{m} \le C^\prime_{A,B} \exp\left((4 - L \log \min\{A,B\}) n\right), \end{aligned} \end{align*}$$

where $C^\prime _{A,B}$ is some constant depending continuously on $A, B$ .

4.2.2 Bounds for the determinant of the moment matrix

Let ${\mathcal A}$ be an $n \times n$ Hermitian matrix, and denote its eigenvalues by $\lambda _n({\mathcal A}) \le \dots \le \lambda _1({\mathcal A})$ . We will use the following result (see, e.g., [Reference Bhatia3, III.2]).

Theorem 4.14 (Weyl’s inequalities)

Let ${\mathcal A}, {\mathcal B}$ be $n \times n$ Hermitian matrices.

For all $k \in \{1, \dots , n\}$ we have

$$\begin{align*}\lambda_k({\mathcal A}) + \lambda_n({\mathcal B}) \le \lambda_{k}({\mathcal A}+{\mathcal B}) \le \lambda_k({\mathcal A}) + \lambda_1({\mathcal B}). \end{align*}$$

We will use the following notation for the entries of moment matrices:

$$\begin{align*}\begin{aligned} \Xi_1(A,k,m; M) & = \sum_{j=0}^M A^{-2j} \binom{j}{k} \binom{j}{m}, \\ \Xi_2(A,B,k,m; M) & = \sum_{j=-M}^{-1} B^{2j} \binom{j}{k} \binom{j}{m} + \sum_{j=0}^M A^{-2j} \binom{j}{k} \binom{j}{m}. \end{aligned} \end{align*}$$

Proposition 4.15. Let $d \in {\mathbb N}$ , $\varepsilon \in (0,1/2)$ , and $A \in \left (1 + \varepsilon , 1/\varepsilon \right )$ . Consider the following (symmetric) matrices

$$\begin{align*}\begin{aligned} {\mathcal M}^{(1)}_d(A) & = \left[ \Xi_1(A,k,m,\infty) \right]_{k,m=0}^{d},\\ {\mathcal M}^{(1)}_d(A;M) & = \left[ \Xi_1(A,k,m,M) \right]_{k,m=0}^{d}. \end{aligned} \end{align*}$$

If $L> e$ is sufficiently large (depending only on $\varepsilon $ ), then

$$\begin{align*}\det {\mathcal M}^{(1)}_d(A; \lfloor Ld \rfloor) \ge 2^{-(d+1)} \det {\mathcal M}^{(1)}_d(A). \end{align*}$$

Remark. Notice that ${\mathcal M}^{(1)}_d(A)$ and ${\mathcal M}^{(1)}_d(A;M)$ are positive-definite matrices. In particular, we may write

$$\begin{align*}{\mathcal M}^{(1)}_d(A)={\mathcal N}_d^t{\mathcal N}_d, \end{align*}$$

where ${\mathcal N}_d$ is the $\infty \times (d+1)$ matrix whose entries are given by $\left [{\mathcal N}_d\right ]_{k,m}=A^{-k}\binom {k}{m}$ . Similarly for ${\mathcal M}^{(1)}_d(A;M)=\left ({\mathcal N}_d(M)\right )^t{\mathcal N}_d(M)$ , where ${\mathcal N}_d(M)$ is the $(M+1)\times (d+1)$ truncation of ${\mathcal M}^{(1)}_d(A)$ . By the Cauchy-Binet formula we have that

$$\begin{align*}\det {\mathcal M}_d^{(1)}(A; M)=\det \left({\mathcal N}_d^t(M){\mathcal N}_d(M)\right)\le \det \left({\mathcal N}_d^t{\mathcal N}_d\right)= \det {\mathcal M}_d^{(1)}(A). \end{align*}$$

Proof. We let ${\mathcal A} = {\mathcal M}_d^{(1)}(A)$ and define the matrix

$$\begin{align*}{\mathcal B} = \left[ \sum_{j> \lfloor Ld \rfloor}^\infty A^{-2j} \binom{j}{k} \binom{j}{m} \right]_{k,m=0}^{d}. \end{align*}$$

Let $\lambda _{d+1}({\mathcal A})$ be the smallest eigenvalue of the positive definite matrix ${\mathcal A}$ . Observe that by Corollary 4.4 and a change of variables (see Section 6), ${\mathcal A}$ is a moment matrix. Thus, by Proposition 4.8, there is a constant $C_1 = C_1(\varepsilon )$ such that $\lambda _{d+1}({\mathcal A}) \ge \exp (-C_1 d)$ . Using Lemma 4.12, we get a pointwise bound for the entries of ${\mathcal B}$ , which in turn bounds its operator norm (up to a factor $d+1$ which can be subsumed into the exponential). Thus, we may choose $L = L(\varepsilon )$ sufficiently large so that for all d large enough $\lambda _1({\mathcal B}) \le \tfrac 12 \lambda _{d+1}({\mathcal A})$ .

Hence, Weyl’s inequalities give

$$ \begin{align*} \det({\mathcal M}_d(A; \lfloor Ld \rfloor)) & = \det({\mathcal A} - {\mathcal B}) = \prod_{j=1}^{d+1} \lambda_j({\mathcal A} - {\mathcal B}) \\ & \ge \prod_{j=1}^{d+1} [ \lambda_j({\mathcal A}) - \lambda_1({\mathcal B}) ] \ge \prod_{j=1}^{d+1} \tfrac12 \lambda_j({\mathcal A}) = 2^{-(d+1)} \det({\mathcal A}). \end{align*} $$

Note that the constant $\tfrac 12$ is arbitrary and can be replaced by (e.g.) $1 - \exp (-d)$ .

Using Lemma 4.13, one can prove the following generalization.

Proposition 4.16. Let $d \in {\mathbb N}$ , $\varepsilon \in (0,1/2)$ , and $A, B \in \left (1 + \varepsilon , 1/\varepsilon \right )$ . Consider the following (symmetric) matrices

$$\begin{align*}\begin{aligned} {\mathcal M}^{(2)}_d(A,B) & = \left[ \Xi_2(A,B,k,m,\infty) \right]_{k,m=0}^{d},\\ {\mathcal M}^{(2)}_d(A,B;M) & = \left[ \Xi_2(A,B,k,m,M) \right]_{k,m=0}^{d}. \end{aligned} \end{align*}$$

If $L> e$ is sufficiently large (depending only on $\varepsilon $ ), then

$$\begin{align*}\det {\mathcal M}^{(2)}_d(A,B; \lfloor Ld \rfloor) \ge 2^{-(d+1)} \det {\mathcal M}^{(2)}_d(A,B). \end{align*}$$

4.3 Tools for the $\boldsymbol{\ell}^{\boldsymbol{2}}$ case

Below we will need an estimate for the number of lattice points inside ellipsoids of a special type.

Proposition 4.17. Let $d \in {\mathbb N}$ and $\gamma , t> 0$ be parameters (that may depend on d), and define the ellipsoid

$$\begin{align*}{\mathcal E} = {\mathcal E}(d; \gamma, t) = \left\{ \textbf{x} = (x_0, \dots, x_d) \in {\mathbb R}^{d+1} \colon \sum_{k=0}^d x_k^2 \gamma^{2k} \le t^2 \right\}. \end{align*}$$

Moreover, let $\Psi $ be a lower unipotent matrix, that is, an $n\times n$ lower triangular matrix, with diagonal elements all equal to $1$ . We denote by $\Psi ({\mathcal E})$ the image of ${\mathcal E}$ under $\Psi $ . Note that this map preserves volume.

For $\gamma \le 1 \le t$ we have

$$\begin{align*}-C_1 (d+1) \le \log \frac{\# \left( \Psi({\mathcal E}) \cap {\mathbb Z}^{d+1} \right)}{\operatorname{\mathrm{Vol}}\left({\mathcal E}\right)} \le C_2 (d+1) \log \min\{ d+1, (1-\gamma)^{-1} \}, \end{align*}$$

where $C_1, C_2>0$ are absolute constants.

Proof. The lower bound (with $C_1 = \log 2$ ) follows from [Reference Gruber and Lekkerkerker9, Sec. 7, Thm. 4].

Let $\mathbf {z}_1, \mathbf {z}_2 \in \Psi ({\mathcal E}) \cap {\mathbb Z}^{d+1}$ be two different lattice points. Notice that by the structure of the matrix $\Psi $ , we have that

$$\begin{align*}\| \mathbf{x}_1 - \mathbf{x}_2 \|_\infty := \| \Psi^{-1}(\mathbf{z}_1) - \Psi^{-1}(\mathbf{z}_2) \|_\infty \ge 1. \end{align*}$$

Now let $\mathbf {x} = (x_0, \dots , x_d) \in {\mathcal E}$ be a preimage of one of the ${\mathbb Z}^{d+1}$ lattice points in $\Psi ({\mathcal E})$ . If $\mathbf {y} = (y_0, \dots , y_d) \in \left [-\tfrac 12, \tfrac 12\right ]^{d+1}$ , then

$$ \begin{align*} \sum_{k=0}^d (x_k + y_k)^2 \gamma^{2k} & = \sum_{k=0}^d x_k^2 \gamma^{2k} + 2 \sum_{k=0}^d x_k y_k \gamma^{2k} + \sum_{k=0}^d y_k^2 \gamma^{2k}\\ & \le 2\left[ t^2 + \tfrac14 \min\{d+1, (1-\gamma^2)^{-1}\} \right], \end{align*} $$

since $2|x_k y_k| \le x_k^2 + y_k^2$ . Therefore, if we denote by ${\mathcal Q}$ the union of all the cubes with side length $1$ centered at preimages of lattice points inside ${\mathcal E}$ , we see that

$$\begin{align*}{\mathcal Q} \subset C_2 (1 + t^{-2} \min\{d+1, (1-\gamma)^{-1}\} {\mathcal E} \subset C_2 \min\{d+1, (1-\gamma)^{-1}\} {\mathcal E}. \end{align*}$$

Since the cubes forming ${\mathcal Q}$ are of volume $1$ and have disjoint interiors, we get the required bound for the number of lattice points.

The following theorem can be proved explicitly using conformal maps, but it also follows as a very special case of [Reference Widom16, Thm. 9.1]. The fine properties of the normalizing constants can be found in [Reference Widom16, Sec. 6].

Theorem 4.18. Let $\eta \in (0,1)$ . Suppose ${\Gamma } = {\mathbb T}_1 \cup {\mathbb T}_2$ is the union of two circles, whose interiors are separated by at least $\eta $ and at most $\frac {1}{\eta }$ , and whose radii are bounded between $\eta $ and $\frac {1}{\eta }$ . Denote by $\left \{q_k\right \}_{k=0}^\infty $ the monic orthogonal polynomial of degree k on ${\Gamma }$ with respect to arc-length, and let

$$\begin{align*}\beta_k^2 = \int_{\Gamma} |q_k|^2 \,\mathrm{d} \sigma, \end{align*}$$

be its normalization constant (squared). The asymptotics of $\beta _k$ are given by

(4.4) $$ \begin{align} \beta_k = \operatorname{\mathrm{Cap}}\left({\Gamma}\right)^k f_k({\Gamma}) (1+o(1)), \qquad k\to\infty, \end{align} $$

where $f_k$ are functions that can be bounded between positive constants depending only on $\eta $ . The implicit error term can also be bounded in terms of $\eta $ only.

We will also require corresponding results for Bergman polynomials (i.e., orthogonal with respect to area measure). The following result closely follows [Reference Gustafsson, Putinar, Saff and Stylianopoulos10, Theorem 4.1]. We give the details in Appendix B.

Theorem 4.19. Let $\eta \in (0,1)$ . Suppose $G = {\mathbb D}_1 \cup {\mathbb D}_2$ is the union of two disks, separated by at least $\eta $ and at most $\frac {1}{\eta }$ , whose radii are bounded between $\eta $ and $\frac {1}{\eta }$ . Let $\{ t_k \}_{k=0}^\infty $ denote the sequence of monic Bergman orthogonal polynomials associated with G, that is

$$\begin{align*}\int_G t_{k_1} (z) t_{k_2}(z) \, \mathrm{d} A(z) = \gamma_{k_1}^2 \cdot \begin{cases} 1, & k_1 = k_2, \\ 0, & k_1 \ne k_2, \end{cases} \end{align*}$$

where $\mathrm {d} A$ denotes the Lebesgue measure on ${\mathbb C}$ . There are constants $0 < C_1(\eta ) < C_2(\eta )$ such that for $k\in {\mathbb N}$

$$\begin{align*}C_1 \frac{\operatorname{\mathrm{Cap}}\left({{\Gamma}}\right)^{k}}{\sqrt{k+1}} \le \gamma_k \le C_2 \frac{\operatorname{\mathrm{Cap}}\left({{\Gamma}}\right)^{k}}{\sqrt{k+1}}. \end{align*}$$

5.1 $\boldsymbol{\ell}^{\boldsymbol{2}}$ volume estimates

We now estimate the volume of the sets ${\mathcal C}^2$ and $\mathcal {D}^2$ . By Corollary 4.3 we have

$$\begin{align*}{\mathcal C}_{d,A,B}^2(t) = \left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \frac{A}{2\pi} \int_{A^{-1} {\mathbb T}} |g_{\textbf{c}}(z)|^2 \mathrm{d} |z| \le t^2, \, \right. \left. \frac{1}{2\pi B} \int_{B {\mathbb T}} |g_{\textbf{c}}(z)|^2 \mathrm{d} |z| \le t^2 \: \right\}, \end{align*}$$

and

$$\begin{align*}\mathcal{D}_{d,A,B}^2(t) = \left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \frac{A^2}{\pi} \int_{A^{-1} {\mathbb D}} |g_{\textbf{c}}(z)|^2 \mathrm{d} A(z) \le t^2,\, \right. \left. \frac{B^2}{\pi} \int_{(B {\mathbb D})^c} |g_{\textbf{c}}(z)|^2 \, \frac{\mathrm{d} A(z)}{|z|^4} \le t^2 \: \right\}. \end{align*}$$

Notice that under a (one-to-one) change of variables $z = \phi (w)$ , the arc-length and area integrals transform as follows,

$$\begin{align*}\int_{\phi({\Gamma})} f(z) \, \mathrm{d} |z| = \int_{\Gamma} f(\phi(w)) |\phi^\prime(w)| \, \mathrm{d} |w|, \qquad \int_{\phi(G)} f(z) \, \mathrm{d} A(z) = \int_G f(\phi(w)) |\phi^\prime(w)|^2 \, \mathrm{d} A(w). \end{align*}$$

We now make the change of variables via the fractional linear transformation,

$$\begin{align*}w = \psi(z) = \frac{z}{1-z}, \quad \phi = \psi^{-1}, \quad z = \phi(w) = \frac{w}{1+w}, \quad \phi^\prime(w) = \frac{1}{(1+w)^2}, \end{align*}$$

such that

$$\begin{align*}h_{\textbf{c}}(w) := (1+w)^{-1} (g_{\textbf{c}} \circ \phi)(w) = \sum_{k=0}^d c_k w^k, \end{align*}$$

and in particular $h_{\textbf {c}}$ is a polynomial. Moreover,

$$\begin{align*}\psi(A^{-1} {\mathbb T}) = {\mathbb T}\left(\frac1{A^2-1},\frac{A}{A^2-1}\right) =: {\mathbb T}_1, \quad \psi(B {\mathbb T}) = {\mathbb T}\left(\frac{B^2}{1-B^2},\frac{B}{B^2-1}\right) =: {\mathbb T}_2. \end{align*}$$

Thus,

$$\begin{align*}{\mathcal C}_{d,A,B}^2(t) = \left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \frac{A}{2\pi} \int_{{\mathbb T}_1} |h_{\textbf{c}}(w)|^2 \mathrm{d} |w| \le t^2, \right. \left. \frac{1}{2\pi B} \int_{{\mathbb T}_2} |h_{\textbf{c}}(w)|^2 \mathrm{d} |w| \le t^2 \right\}, \end{align*}$$

and

$$\begin{align*}\mathcal{D}_{d,A,B}^2(t) = \left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \frac{A^2}{\pi} \int_{{\mathbb D}_1} |h_{\textbf{c}}(w)|^2 \frac{\mathrm{d} A(w)}{|1+w|^{2}} \le t^2, \right. \left. \frac{B^2}{\pi} \int_{{\mathbb D}_2} |h_{\textbf{c}}(w)|^2 \frac{\mathrm{d} A(w)}{|w|^{4} |1+w|^{-2}} \le t^2 \right\}, \end{align*}$$

where

$$\begin{align*}{\mathbb D}_1 := {\mathbb D}\left(\frac1{A^2-1},\frac{A}{A^2-1}\right), \quad {\mathbb D}_2 := {\mathbb D}\left(\frac{B^2}{1-B^2},\frac{B}{B^2-1}\right). \end{align*}$$

5.1.1 Volume of $\boldsymbol{\mathcal{C}}^{\boldsymbol{2}}$

Let $\{ q_k(w) \}_{k=0}^\infty $ be the sequence of monic orthogonal polynomials with respect to arc-length measure on the curve $\Gamma ~:= ~{\mathbb T}_1~\cup ~{\mathbb T}_2$ in increasing degree. Then, we can write

$$\begin{align*}h_{\mathbf{c}}(w) = \sum_{k=0}^d \alpha_k(\mathbf{c}) q_k(w), \quad \text{where} \quad \alpha_d = c_d, \qquad \beta_k^2 := \int_\Gamma |q_k(w)|^2 \mathrm{d} |w|. \end{align*}$$

Furthermore,

(5.1) $$ \begin{align} {\mathcal E}(\beta_0,\dots,\beta_d; t_1) \subset {\mathcal C}_{d,A,B}^2(t) \subset {\mathcal E}(\beta_0,\dots,\beta_d; t_2), \end{align} $$

where ${\mathcal E}(\beta _0,\dots ,\beta _d;t)$ is the ellipse

$$\begin{align*}{\mathcal E}(\beta_0,\dots,\beta_d;t) = \left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \sum_{k=0}^d |\alpha_k(\textbf{c})|^2 \beta_k^2 \le t^2 \right\}, \end{align*}$$

and

$$\begin{align*}t_1 := \sqrt{2 \pi \min\{A^{-1}, B\}} \, t, \qquad t_2 := \sqrt{2 \pi (A^{-1} +B)} \, t. \end{align*}$$

Let $\eta \in (0,1)$ , and suppose $A,B> 1$ satisfy the conditions of Theorem 4.18. In particular, A and B are bounded away from $1$ and $\infty $ in terms of $\eta $ . Hence, the constants $\{ \beta _k \}_{k\in {\mathbb N}}$ satisfy

$$\begin{align*}\beta_k = \operatorname{\mathrm{Cap}}\left({\Gamma}\right)^k g_k(A,B,d), \end{align*}$$

where $g_k$ are functions that are bounded between positive constants depending only on $\eta $ . Therefore, by (5.1),

$$\begin{align*}\left[ C_1(\eta) t_1 \right]^{d+1} \le \frac{\operatorname{\mathrm{Vol}}\left({\mathcal C}_{d,A,B}^2(t)\right)}{\operatorname{\mathrm{Vol}}\left(B^{d+1}_2\right) \operatorname{\mathrm{Cap}}\left({\Gamma}\right)^{-d(d+1)/2}} \le \left[ C_2(\eta) t_2 \right]^{d+1}. \end{align*}$$

This proves the $\ell ^2$ volume asymptotics in the first part of Theorem 2.1. The second part of the theorem follows by an application of Proposition 4.17, with the ellipsoid $\Psi ^{-1}\left ({\mathcal E}(\beta _0,\dots ,\beta _d; t_\ell )\right )$ , where $\Psi $ is the change of basis given by the $\alpha _k$ s.

Now, notice that the leading coefficient of $h_{\mathbf {c}}$ is a nonzero integer. Hence, it follows that there is a constant $C_\eta> 0$ such that

$$\begin{align*}\int_{{\mathbb T}_1} |h_{\textbf{c}}(w)|^2 \mathrm{d} |w| \ge C_\eta \quad \text{or} \quad \int_{{\mathbb T}_2} |h_{\textbf{c}}(w)|^2 \mathrm{d} |w| \ge C_\eta. \end{align*}$$

If we choose t sufficiently small, depending on $\eta $ , then we conclude that there are no integer lattice points inside ${\mathcal C}_{d,A,B}^2(t)$ . This proves the first part of Theorem 2.3.

5.1.2 Volume of $\boldsymbol{\mathcal {D}}^{\boldsymbol{2}}$

For Bergman orthogonal polynomials notice that first we can bound the weights $|1+w|^{-2}$ on ${\mathbb D}_1$ , and $|w|^{-4}|1+w|^2$ on ${\mathbb D}_2$ from above and from below in terms of $\eta $ . Thus, there are constants $c^{(j)} = c^{(j)}(\eta ), \,j\in \{1,2\},$ such that

$$\begin{align*}\begin{aligned} \mathcal{D}_{d,A,B}^2(t) & \subset \left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon c^{(1)} \int_{{\mathbb D}_1} |h_{\textbf{c}}(w)|^2 \mathrm{d} A(w) \le t^2, \quad c^{(2)} \int_{{\mathbb D}_2} |h_{\textbf{c}}(w)|^2 \mathrm{d} A(w) \le t^2 \right\}\\ & \subset \left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \tfrac12 \min\{c^{(1)},c^{(2)}\} \left( \int_{{\mathbb D}_1} + \int_{{\mathbb D}_2} \right) |h_{\textbf{c}}(w)|^2 \mathrm{d} A(w) \le t^2 \right\}, \end{aligned} \end{align*}$$

and similarly in the other direction with a different constant.

Now, let $t_k$ be the Bergman monic orthogonal polynomials with respect to area measure on ${\mathbb D}_1 \cup {\mathbb D}_2$ , and let $\gamma _k$ be their normalizing constants. By Theorem 4.19 we have

$$\begin{align*}C_1(\eta) \frac{\operatorname{\mathrm{Cap}}\left({\Gamma}\right)^{k+1}}{\sqrt{k+1}} \le \gamma_k \le C_2(\eta) \frac{\operatorname{\mathrm{Cap}}\left({\Gamma}\right)^{k+1}}{\sqrt{k+1}}, \end{align*}$$

which, together with the previous discussion, gives

$$\begin{align*}\left( \widetilde{C_1}(\eta) t \right)^{d+1} \le \frac{\operatorname{\mathrm{Vol}}\left(\mathcal{D}_{d,A,B}^2(t)\right)}{\operatorname{\mathrm{Cap}}\left({\Gamma}\right)^{-d(d+1)/2}} \le \left( \widetilde{C_2}(\eta) t \right)^{d+1}. \end{align*}$$

Here we used the asymptotics of the volume of the Euclidean ball

$$\begin{align*}\operatorname{\mathrm{Vol}}\left(B^{d+1}_2\right) = \left(\frac{2\pi e (1+o(1))}{d+1}\right)^{(d+1)/2}, \quad \text{as} \quad d \to \infty. \end{align*}$$

Hence, choosing t sufficiently large, the volume of $\mathcal {D}^2$ is exponentially large in d (as long as ${\mathrm {Cap}}\left ({\Gamma }\right ) \le 1$ ). Applying [Reference Gruber and Lekkerkerker9, Sec. 7.2, Thm. 1] proves the second part of Theorem 2.3.

5.2 Refined estimate for the critical growth constraint on the natural numbers

By essentially the same argument as in Section 5.1 we find

$$ \begin{align*} {\mathcal C}_{d,A}^2(t) & = \left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \sum_{n\in{\mathbb N}} |P_{\textbf{c}}(n) A^{-n}|^2 \le t^2 \right\} = \left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \frac{A}{2\pi} \int_{{\mathbb T}_1} |h_{\textbf{c}}(w)|^2 \mathrm{d} |w| \le t^2 \right\} \\ & = \left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \frac{A}{2\pi} \sum_{k=0}^d |\alpha_k(\textbf{c})|^2 \beta_k^2 \le t^2 \right\}, \end{align*} $$

where as before

$$\begin{align*}{\mathbb T}_1 = {\mathbb T}\left(\frac1{A^2-1},\frac{A}{A^2-1}\right), \qquad h_{\textbf{c}}(w) = \sum_{k=0}^d c_k w^k. \end{align*}$$

Here the monic orthogonal polynomials with respect to arc-length on ${\mathbb T}_1$ and their normalization constants are given explicitly by

$$\begin{align*}p_k(w) = (w - (A^2-1)^{-1})^k,\qquad \beta_k^2 = 2 \pi \left( \frac{A}{A^2-1} \right)^{2k+1}. \end{align*}$$

Thus, taking the prefactor $A/(2\pi )$ into account, we have

$$\begin{align*}\operatorname{\mathrm{Vol}}\left({\mathcal C}_{d,A}^2(t)\right) = \left( \frac{A^2-1}{A} \right)^{(d+1)^2/2} \operatorname{\mathrm{Vol}}\left(B^{d+1}_2\right) \left( \frac{t}{\sqrt{A}}\right)^{d+1} = \frac{(A^2-1)^{(d+1)^2/2}}{A^{(d+1)(d+2)/2}} \operatorname{\mathrm{Vol}}\left(B^{d+1}_2\right) t^{d+1}. \end{align*}$$

Evidently if $c_k$ is the largest nonzero coefficient of $h_{\textbf {c}}$ , then $\int _{{\mathbb T}_1} |h_{\textbf {c}}(w)|^2 \mathrm {d} |w| \ge c_k^2 \beta _k^2$ . Hence, when $A = \varphi $ and $t < \sqrt {\varphi }$ there are no integer-valued polynomials P satisfying

$$\begin{align*}\sum_{n\in{\mathbb N}} |P_{\textbf{c}}(n) \varphi^{-n}|^2 \le t^2. \end{align*}$$

This proves the first part of Theorem 1.2. We now turn to the proof of the second part.

We will use the results from [Reference Suetin14, Chap. III]. We denote by $\mathrm {S}.x$ and $\mathrm {S}.x.y$ theorem and equation numbers from this paper.

Denote

$$ \begin{align*} \mathcal{D}_{d,\varphi}^2(t) & = \left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \sum_{n\in{\mathbb N}} \frac{1}{n+1} |P_{\textbf{c}}(n) \varphi^{-n}|^2 \le t^2 \right\} \\ & = \left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \frac{\varphi^2}{\pi} \int_{{\mathbb D}_1} |h_{\textbf{c}}(w)|^2 \frac{\mathrm{d} A(w)}{|1+w|^2} \le t^2 \right\}\\ & = \left\{ \textbf{c} \in {\mathbb R}^{d+1} \colon \frac{\varphi^2}{\pi} \sum_{k=0}^d |\widetilde{\alpha}_k(\textbf{c})|^2 \gamma_k^2 \le t^2 \right\}, \end{align*} $$

where now $\gamma _k$ are the normalizing constants of monic orthogonal polynomials with respect to the weight $\frac {\mathrm {d} {\mathbb A}(w)}{|1+w|^2}$ on ${\mathbb D}_1 = {\mathbb D}(\varphi ^{-1} , 1)$ , and $\widetilde {\alpha }_k(\textbf {c})$ are the coefficients after the change of basis.

For fixed $\varepsilon \in (0,1)$ , we will find a condition on $t = t(\varepsilon )$ so that the volume of $\mathcal {D}_{d,\varphi }^2(t)$ is at least $(2+\varepsilon )^{d+1}$ . Then an application of [Reference Gruber and Lekkerkerker9, Sec. 7.2, Thm. 1] proves there are infinitely many integer-valued polynomials P satisfying the growth condition

$$\begin{align*}\sum_{n\in{\mathbb N}} \frac{1}{n+1} |P_{\textbf{c}}(n) \varphi^{-n}|^2 \le t^2. \end{align*}$$

The asymptotics of $\gamma _k$ follow from Theorem $\mathrm {S}.3.1$ (in Suetin’s notation it is equal to $\lambda _k^{-1}$ ). We introduce some notation required for the statement of the results.

Let $\Phi $ be the conformal map from the exterior of ${\mathbb D}_1$ to the exterior of the unit disk ${\mathbb D}$ , satisfying $\Phi (\infty ) = \infty $ and $\Phi ^\prime (\infty )> 0$ , that is, the map $\Phi (z) = z - \varphi ^{-1}$ . We need to introduce an analytic function $D: {\mathbb D}^c \to {\mathbb D}_1^c$ such that

$$\begin{align*}\left|D(w)\right|{}^2 = \frac{1}{\left|1+\left(w + \varphi^{-1} \right)\right|{}^2} = \frac{1}{\left|w + \varphi\right|{}^2} \quad \text{on} \quad |w| = 1. \end{align*}$$

Explicitly it is given by the following expression

$$\begin{align*}D(w) = \exp \left( -\frac{1}{4 \pi} \int_0^{2\pi} \log \frac{1}{| e^{i \theta} + \varphi|^2} \frac{e^{i\theta} + w}{e^{i \theta} - w} \, d\theta \right). \end{align*}$$

Using Suetin’s notation, let

$$\begin{align*}g(z) = \frac{\Phi^\prime(z)}{D\left(\Phi(z)\right)} = \frac{\Phi^\prime(z)}{D\left( z - \varphi^{-1} \right)} = \frac{1}{D(z - \varphi^{-1})}, \quad \text{and} \quad \alpha_0 = g(\infty) = \frac{1}{D(\infty)}. \end{align*}$$

Since the function $z + \varphi $ does not vanish in ${\mathbb D}$ , the mean value theorem for harmonic functions gives

$$\begin{align*}\log D(\infty) = \frac{1}{4 \pi} \int_0^{2\pi} \log \frac{1}{| e^{i \theta} + \varphi|^2} \, d\theta = - \frac{1}{2 \pi} \int_0^{2\pi} \log | e^{i \theta} + \varphi| \, d\theta = - \log \varphi. \end{align*}$$

Now by Theorem $\mathrm {S}.3.1$ we have

$$\begin{align*}\gamma_k^2 = \frac{\pi}{\alpha_0^2(k+1)} \left(1 + o(1)\right) = \frac{\pi}{\varphi^2(k+1)} \left(1 + o(1)\right). \end{align*}$$

By the asymptotic formula for the volume of the Euclidean ball and Stirling’s approximation, we have

$$\begin{align*}\operatorname{\mathrm{Vol}}\left(\mathcal{D}_{d,\varphi}^2(t)\right) = \operatorname{\mathrm{Vol}}\left(B_2^{d+1}\right) \prod_{k=0}^{d} \gamma_k^{-1} \left(\frac{\sqrt{\pi t}}{\varphi} \right)^{d+1} = (\sqrt{2\pi})^{d+1} t^{d+1} \cdot \exp(o(1)). \end{align*}$$

Finally, we conclude that if $t \ge \sqrt {2/\pi } + \varepsilon $ , then the required condition is satisfied.