Lemma 4.6 — $\|\mathcal E\|<2.01\,e^{-\pi\alpha^2\theta/2}<2^{-\alpha^2\theta}$

Statement. For all natural numbers $s, t$ with $s \ne 0$, $t \ne 0$, $s < t$ and $\gcd(s,t) = 1$, and for every real $\alpha > 0$ satisfying

both of the following strict inequalities hold:

Here $2.01$ is the exact rational $201/100$, $\theta$ is computed in the reals (real division $t/s$), and $2^{-\alpha^2\theta}$ is the real power with positive base. The hypotheses $s \ne 0$, $t \ne 0$ enter as typeclass assumptions; $t \ne 0$ is in any case implied by $s < t$. Nothing else is assumed: in particular $s = 1$ is allowed (then coprimality is automatic and $t \ge 2$ is arbitrary), and no parity or size condition is placed on $s$ or $t$. Since $s < t$, $\theta > 0$, so the hypothesis $\alpha^2\theta \ge 1$ is satisfiable (e.g. by any sufficiently large $\alpha$); the hypotheses are not vacuous.

The space and the norm. Vectors in $\mathbb{C}^s$ are functions $u : \mathbb{Z}/s\mathbb{Z} \to \mathbb{C}$, indexed by residues $\ell \in \mathbb{Z}/s\mathbb{Z}$; each residue is identified with its canonical representative in $\{0, 1, \dots, s-1\}$ when an integer or real value of the index is needed. Likewise for $\mathbb{C}^t$ with $\mathbb{Z}/t\mathbb{Z}$. The vector norm is the supremum norm $\|u\| = \max_{\ell} |u_\ell|$ (complex modulus), and $\|\mathcal{E}\|$ is the operator norm of the $\mathbb{C}$-linear map $\mathcal{E} : \mathbb{C}^s \to \mathbb{C}^s$ with respect to the sup norm on both sides, i.e. $\|\mathcal{E}\| = \inf\{c \ge 0 : \|\mathcal{E}u\| \le c\|u\| \ \forall u\}$, which for this norm equals the maximum absolute row sum $\max_{\ell} \sum_{r} |\mathcal{E}_{\ell r}|$ of its $s \times s$ matrix.

Unfolding $\mathcal{E}$. By definition $\mathcal{E} = \mathcal{N} - \mathcal{I}$, where $\mathcal{I}$ is the identity on $\mathbb{C}^s$ and $\mathcal{N} = \mathcal{C} \circ \mathcal{T} \circ \mathcal{D}$ is the composite of three maps:

• Nearest integer and $\beta$. For real $x$, $[x] := \lfloor x + \tfrac12 \rfloor$ (nearest integer, ties rounded upward). For an integer $\ell$, $\beta_\ell := \dfrac{t\ell}{s} - \Big[\dfrac{t\ell}{s}\Big] \in [-\tfrac12, \tfrac12)$.
• Diagonal map $\mathcal{D} : \mathbb{C}^s \to \mathbb{C}^s$: $(\mathcal{D}u)_r = e^{\pi \alpha^2 \beta_r^2}\, u_r$ for $r \in \{0,\dots,s-1\}$.
• Resampling map $\mathcal{T} : \mathbb{C}^s \to \mathbb{C}^t$: for $k \in \{0,\dots,t-1\}$,

which is the same as $\sum_{j \in \mathbb{Z}} e^{-\pi\alpha^2 t^2 (k/t - j/s)^2} v_{j \bmod s}$. (The inner series is an unconditional sum over $\mathbb{Z}$, assigned the value $0$ if not summable; for $\alpha > 0$ it is a convergent Gaussian series, so this junk convention plays no role.)

• Row-deleting map $\mathcal{C} : \mathbb{C}^t \to \mathbb{C}^s$: for $\ell \in \{0,\dots,s-1\}$, $(\mathcal{C}w)_\ell = w_{\kappa_\ell}$, where $\kappa_\ell \in \{0,\dots,t-1\}$ is the residue of the integer $[t\ell/s]$ modulo $t$. Note: since $0 \le t\ell/s < t$, one has $[t\ell/s] \in \{0,\dots,t\}$, and the value $t$ (which occurs when $t\ell/s \ge t - \tfrac12$) is reduced to $\kappa_\ell = 0$; then the row index fed into $\mathcal{T}$ is $k = 0$, not $k = t$.

Composing, $\mathcal{E}$ is the $s \times s$ matrix with entries ($\ell, r \in \{0,\dots,s-1\}$)

with $\delta_{\ell r}$ the Kronecker delta, and claim (i) says $\max_\ell \sum_r |\mathcal{E}_{\ell r}| < 2.01\, e^{-\pi\alpha^2\theta/2}$. The definition file also defines maps $\mathcal{S}$, $\mathcal{P}_s$, $\mathcal{P}_t$ and the DFT, but none of them enters this theorem.

Remarks on the two conjuncts. The theorem is a conjunction; both parts are asserted under the same hypotheses. Conjunct (ii) does not involve $\mathcal{E}$ at all: it depends on $s, t, \alpha$ only through the single real number $x = \alpha^2\theta$, and asserts $2.01\, e^{-\pi x/2} < 2^{-x}$ for that $x$ (which satisfies $x \ge 1$ by hypothesis). Coprimality of $s$ and $t$ and the strict inequality $s < t$ are hypotheses of the whole statement, and are not otherwise encoded in the definitions of $\mathcal{E}$ or $\theta$ (which make sense for any nonzero $s, t$).