TAOCP 3.3.4 Exercise 13

Section 3.3.4: The Spectral Test

Exercise 13. [**] $[HM22]$ Lemma A uses the fact that $U$ is nonsingular to prove that a positive definite quadratic form attains a definite, nonzero minimum value at nonzero integer points. Show that the same result is necessary: by exhibiting a quadratic form (19) whose matrix of coefficients is singular, and for which the values of $f(x_1, \ldots, x_t)$ get arbitrarily near zero (but never reach it) at nonzero integer points $(x_1, \ldots, x_t)$.

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Let

$$ f(x_1,\ldots,x_t)=\sum_{i,j}u_{ij}x_ix_j $$

be a quadratic form. We must exhibit a form whose coefficient matrix is singular, such that

$$ f(x_1,\ldots,x_t)>0 $$

for every nonzero integer point, yet the values of $f$ at nonzero integer points can be made arbitrarily small.

Consider the two-variable quadratic form

$$ f(x,y)=(x-\alpha y)^2, $$

where $\alpha$ is an irrational number.

Its coefficient matrix is

$$ U= \begin{pmatrix} 1 & -\alpha\ -\alpha & \alpha^2 \end{pmatrix}. $$

Since

$$ \det U=\alpha^2-\alpha^2=0, $$

the matrix $U$ is singular.

Now let $(x,y)$ be a nonzero integer point. If $f(x,y)=0$, then

$$ x-\alpha y=0, $$

hence $x=\alpha y$.

If $y\neq0$, this would imply

$$ \alpha=\frac{x}{y}, $$

contradicting the irrationality of $\alpha$. If $y=0$, then $x=0$ as well. Therefore

$$ f(x,y)>0 $$

for every nonzero integer pair $(x,y)$. Thus the value $0$ is never attained at a nonzero integer point.

It remains to show that the values can be arbitrarily close to $0$. By the theory of continued fractions, an irrational number $\alpha$ has infinitely many rational approximations $p/q$ satisfying

$$ \left|\alpha-\frac pq\right|<\frac1{q^2}. $$

Taking the integer point $(p,q)$, we obtain

$$ f(p,q) =(p-\alpha q)^2 =q^2\left(\frac pq-\alpha\right)^2 <q^2\left(\frac1{q^2}\right)^2 =\frac1{q^2}. $$

Since there are infinitely many such pairs $(p,q)$ with $q\to\infty$,

$$ f(p,q)<\frac1{q^2}\longrightarrow 0. $$

Hence the values of $f$ at nonzero integer points become arbitrarily small, but they never equal $0$.

This provides the required example. Therefore the nonsingularity hypothesis in Lemma A is indeed necessary. ∎