Here, we are going to prove Theorem 1.14 from the introduction. Therefore, fix a r× s matrix A. We first need a technical lemma.
Lemma 4.1. If Au| ∈ Fq[X]r for some u|̸= 0, then A is not badly approximable.
Proof. Assume that A is badly approximable. Let us fix some notation. First, set
A =
which is in Fq[X]r. Then, by Dirichlet’s theorem,
|Q1f11+ Q2f21+· · · + Qrfr1− P1| < q−⌊s−1N r⌋
|Q1f12+ Q2f22+· · · + Qrfr2− P2| < q−⌊sN r−1⌋ ...
|Q1f1s−1+ Q2f2s−1+· · · + Qrfrs−1− Ps−1| < q−⌊sN r−1⌋
has infinitely many solutions in Q1, Q2, . . . , Qr and P1, P2, . . . , Ps−1 with N = max1≤i≤rdeg(Qi). Next, multiply both sides of the above inequality by |Us| and set Q′i := UsQi and Pj′ := UsPj for all i, j. Then, we obtain
|Q′1f11+ Q′2f21+· · · + Q′rfr1− P1′| < |Us|q−⌊sN r−1⌋
|Q′1f12+ Q′2f22+· · · + Q′rfr2− P2′| < |Us|q−⌊s−1N r⌋ ...
|Q′1f1s−1+ Q′2f2s−1+· · · + Q′rfrs−1− Ps′−1| < |Us|q−⌊sN r−1⌋. This implies that
|Q′1f11+ Q′2f21+· · · + Q′rfr1 − P1′| < q−⌊N ′rs−1⌋−c1
|Q′1f12+ Q′2f22+· · · + Q′rfr2 − P2′| < q−⌊N ′rs−1⌋−c1 ...
|Q′1f1s−1+ Q′2f2s−1+· · · + Q′rfrs−1− Ps′−1| < q−⌊N ′rs−1⌋−c1
(4.1)
has infinitely many solutions in Q′1, . . . , Q′r and P1′, . . . , Pr′, where N′ = maxi=1,...,rdeg(Q′i) and c1 is a suitable constant. Now, consider
Usf1sQ′1+· · · + UsfrsQ′r
=
∑r i=1
(Ri− U1fi1− · · · − Us−1fis−1) Q′i
=
∑r i=1
Q′iRi−
s−1
∑
j=1
Uj(
Q′1f1j +· · · + Q′rfrj − Pj′
)−
s−1
∑
j=1
UjPj′
This implies that
∑r i=1
UsfisQ′i+
s−1
∑
j=1
UjPj′−
∑r i=1
Q′iRi =−
s−1
∑
j=1
Uj(
Q′1f1j +· · · + Q′rfrj − Pj′
)
Hence,
∑r i=1
UsfisQ′i+
s−1
∑
j=1
UjPj′ −
∑r i=1
Q′iRi
≤ max
j=1,...,s−1
{|Uj| Q′1f1j +· · · + Q′rfrj− Pj′ } < q−⌊N ′rs−1⌋−c2, where c2 is a suitable constant. Dividing both sides by|Us| gives
∑r i=1
fisQ′i+
∑s
j=1UjPj′−∑r
i=1Q′iRi Us
< q−⌊s−1N ′r⌋−c3,
where c3 is a suitable constant. Since Us divides Q′i and Pj′ for all i, j, we obtain T =
∑s
j=1UjPj′−∑r
i=1Q′iRi Us
is a polynomial. Thus, we have proved that
|Q′1f1s+· · · + Q′rfrs+ T| < q−⌊sN r−1⌋−c3.
Now, set q′ = [Q′1, Q′2, . . . , Q′r] and c = min{c1, c3}. By the above inequality and (4.1),
∥{q′A}∥ < q−⌊N ′rs−1⌋−c
has infinitely many solutions. Consequently, we obtain that A is not badly ap-proximable, a contradiction. Hence, the proof is finished.
For the next five lemmas, we assume that A = [fij]r×s is badly approx-imable. Then, there exists a constant c > 0 such that for all q ∈ Fq[X]r, q̸= 0 with deg(q) = n,
∥{qA}∥ > 1 q⌊nrs ⌋+c. Lemma 4.2. {{qA} : q ∈ Fq[X]r} is dense in Ls.
Proof. Let us fix n ∈ N and g = [g1, g2, . . . , gs]∈ Ls, where gj = gj(1)X−1+ gj(2)X−2+· · · , ∀j = 1, 2, . . . , s.
We have to show that there exists q = [Q1, Q2, . . . , Qr] with deg(Qi) = Ni such that
∥{qA} − g∥ < 1
qn. (4.2)
First, for i = 1, 2, . . . , r and j = 1, 2, . . . , s, set
Then, the inequality (4.2) has a solution if and only if aA′ = b has a solution a.
In order to prove that this system is solvable, we have to show that rank(A′) = sn as N is large enough. Assume that there exist α1, . . . , αsn not all zero such that Hence, (4.3) can be rewritten to
| {P1fi1+· · · + Psfis} | < q−N−1 for all i = 1, 2, . . . , r. This implies that
∥ {Ap|} ∥ < q−N−1.
On the other hand, since A is badly approximable, Lemma 4.1 implies that Ap| does not belong toFq[X]r,∀p ̸= 0. Consequently, since the number of p is finite (since n is fixed),
deg(p)<n, pmin ̸=0∥ {Ap|} ∥ ≥ q−N−1,
for N large enough, a contradiction. Hence, we obtain that p = 0 which implies α1 = α2 = · · · = αsn = 0. Thus, our claimed result is proved. Therefore, there exists a solution of aA′ = b. This implies that for all n ∈ N, (4.2) has a solution.
Finally, we have proved that {{qA} : q ∈ Fq[X]r} is dense in L.
The next two lemmas are proved as in the last chapter. Consequently, we will omit the proofs.
Lemma 4.3 (0-1 law). Let a measurable set E inLsbe invariant under the action
· + {qA} for all q ∈ Fq[X]r. Then, we have m(E) = 0 or 1.
Lemma 4.4. Let E :=
{
g :∥ {qA} − g∥ < 1
q⌊nrs⌋+ln with n = deg(q) has infinitely many solutions }
. Then, E is invariant under the action · + {qA} for all q ∈ Fq[X]r and hence m(E) = 0 or 1.
Next, we need the following result which is similar to Lemma 3.4 from the last chapter.
Lemma 4.5. Let g∈ Ls and d > 0. Then, the number of{qA} with deg(q) ≤ N belonging to B(
g, q−d)
is at most max{qN r+cs−ds, 1}.
Proof. For the proof, we use the second method of proof of Lemma 3.4. Therefore, fix q, q′ ∈ Fq[X]r with deg(q), deg(q′)≤ N. Since A is badly approximable, we have
∥{qA} − {q′A}∥ = ∥{(q − q′)A}∥ > 1
q⌊deg(qs−q′)r⌋+c ≥ 1 q⌊N rs ⌋+c.
This means that the distance between any two points {qA} and {q′A} is more than q−⌊N rs ⌋−c. Then, we consider the following two cases.
1. If q−⌊N rs ⌋−c ≥ q−d, then there is at most one point in B(g, q−d).
2. If q−⌊N rs ⌋−c < q−d, then the number of points in B(g, q−d) is at most (q−d)s
(
q−⌊N rs ⌋−c
)s ≤ qN r+cs−ds.
Hence, our claim is proved.
Lemma 4.6. Let ln be a sequence with ∑ incorrect. Hence, there exists k0 ∈ N such that
m We first estimate the number of elements of LN. Let
N∪−1
are disjoint∀i. By (4.5), we get 1
Using Lemma 4.5, the number of q with deg(q)≤ N such that {qA} belongs to Then, we obtain
∥{q1A} − {q2A}∥ < 1 q⌊N rs ⌋+l′N. By Lemma 4.5, the number of {qA} belonging to B(
{q1A}, q−⌊N rs ⌋−l′N) is at most max{qN r−Nr−sl′N+cs, 1} = max{q−sl′N+cs, 1} = 1. Thus, we get {q1A} = {q2A}, a contradiction. Consequently, (4.6) holds. Now, we show that any two balls appearing on the left side of (4.6) are disjoint. We again use proof by contradiction. Suppose there are q1, q2 ∈ LN such that B(
{q1A}, q−⌊N rs ⌋−l′N) and B(
{q2A}, q−⌊N rs ⌋−l′N)
are not disjoint. Thus, we know that these two balls are equal. This implies that
∥{q1A} − {q2A}∥ = ∥{(q1− q2)A}∥ < 1 By Lemma 4.5 again, the number of {qA} belonging to B(
0, q−⌊N rs ⌋−l′N) is at most max{qcs−sl′N, 1} = 1. Consequently, {q1A} = {q2A}, a contradiction. By
the latter claim and (4.6), we obtain
q−sln′ diverges, we have a contradiction for N large enough.
Now, we consider the case q = 2 and r = 1. Since∑
n≥0q−sl′n =∞, we have either
∑
n≥0q−sl′2n =∞ or ∑
n≥0q−sl′2n+1 =∞. Without loss of generality, assume that the first case holds. Then, the same proof as above can be used with the only difference that instead of LN, we consider
L2N := Hence, we obtain
m for N large enough.
Proposition 4.1.
Sr,s ⊇{
A∈ Lr×s : A is badly approximable}
Proof. Let A be badly approximable. Then, we have to show that
and Lemma 4.4 implies our claim.
Proposition 4.2.
Sr,s ⊆{
A∈ Lr×s : A is badly approximable}
Proof. Assume that A is not badly approximable. We will show that we can choose a sequence ln such that ∑∞
n=1q−sln = ∞ but for almost every g ∈ Ls, there are finitely many q with
∥{qA} − g∥ < 1
q⌊nrs⌋+ln, q∈ Fq[X]r, deg(q) = n.
Since A is not badly approximable, there exists a sequence q(i) = [ Then, we have
∑∞
On the other hand, assume without loss of generality that qni =∥q(i)∥ = |Q(i)1 |.
We have to show that
∪
ti−1≤n<ti
∪
deg(q)=n
B(
{qA}, q−⌊nrs⌋−ln)
⊂∪ B
({q′A}, q−⌊rtis ⌋+2) ,
where the right union runs over all q′ = [Q′1, Q′2, . . . , Q′r] which fulfil the conditions
|Q′1| ≤ qni−1, |Q′2| ≤ qti−1, . . . , |Q′r| ≤ qti−1.
Fix {qA} with ti−1 ≤ deg(q) = n < ti. Then, there exists a polynomial h such that |Q1+ hQ(i)1 | ≤ qni−1. Note that |h| ≤ qti−1−ni. Now set
q′ = [Q1 + hQ(i)1 , Q2+ hQ(i)2 , . . . , Qr+ hQ(i)r ].
Then, we obtain
∥{qA} − {q′A}∥ ≤ |h|∥{q(i)A}∥ < qti−1−niq−⌊rtis ⌋−i= q−⌊rtis ⌋−1. Note that
q−⌊nrs ⌋−ln = q−⌊nrs ⌋−⌊r(ti−n)s ⌋ < q−nrs −r(ti−n)s +2 ≤ q−⌊rtis ⌋+2. Hence, we have
B(
{qA}, q−⌊nrs ⌋−ln)
⊂ B(
{q′A}, q−⌊rtis ⌋+2) .
Therefore, our claim is proved. Now, we estimate the measure of union of these balls
m
∪
ti−1≤n<ti
∪
deg(q)=n
B (
{qA}, 1 q⌊nrs ⌋+ln
) ≤ qs(−⌊rtis ⌋+2)qni+ti(r−1) ≤ q3s−i.
Consequently,
∑∞ i=1
m
∪
ti−1≤n<ti
∪
deg(q)=n
B (
{qA}, 1 q⌊nrs ⌋+ln
) ≤∑∞
i=1
q3s−i <∞.
Hence, for almost every g ∈ Ls, there are finitely many q’s with deg(q) = n satisfying ∥{qA} − g∥ < q−⌊nrs ⌋−ln.
Finally, Proposition 4.1 and Proposition 4.2 imply Theorem 1.14.
Chapter 5 Conclusion
We conclude this thesis with some remarks.
First, note that in the real number case, the approximation function in Kurzweil’s theorem is assumed to be monotonic (compare with the theorems of Kristensen in Section 1.2 were also some monotonicity assumptions are used). In this work, on the other hand, the approximation function is of the form q−n−ln with no monotonicity assumptions on ln. However, note that our approximation function tends to 0 as n tends to infinity (this was not assumed by Kurzweil).
In fact, we guess that if one replaces our approximation function by q−ln, then in order for the result to hold, a monotonicity condition on ln similar to the one used by Kurzweil is needed.
Next, we briefly discuss possible almost sure results for the number of solutions of (1.6) (similar considerations can be made for the higher dimensional case). Let us denote a sequence of random variables counting solutions of (1.6) by
XN := #{solutions in Q with Q monic of (1.6) with n ≤ N}.
Then, we have that
XN = ∑
n≤N
∑
deg(Q)=n
χB({Qf},q−n−ln),
where χA denotes the indicator function of the set A. Hence, we obtain for the
expected value
E [XN] = ∑
n≤N
∑
deg(Q)=n
q−n−ln = ∑
n≤N
q−ln.
An interesting question is whether or not one can prove a strong law of large numbers for the number of solutions? More precisely, is it true that for any badly approximable f , we have
XN ∼ ∑
n≤N
q−ln a.s. ?
If yes, what can be said about the error term (which should then depend on Diophantine approximation properties of f )?
Overall, there are still interesting questions left concerning inhomogeneous Diophantine approximation in the field of formal Laurent series.
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