In this seminar, I’m going to be discussing joint work with Kyle Hambrook.
This work is still in progress, so be aware that it may still contain errors at this
point.
A set E Ă Rn is said to have Fourier dimension at least s if the following
condition holds: there exists a probability measure µ supported on E with the
property that, for every ǫ ą 0, there exists a constant Cǫ such that
µ
ppξq ď Cǫ |ξ|´s{2`ǫ .
(If this holds “on average”, then E has Hausdorff dimension at least s. So
the Fourier dimension of E is no more than the Hausdorff dimension of E).
Note that the Fourier dimension of E depends on the ambient space.
A set E is called a Salem set if the Fourier dimension of E is equal to the
Hausdorff dimension of E.
Example: The sphere in Rn is a Salem set.
Nonexamples: Hyperplanes in Rn have Fourier dimension 0. The Cantor
middle-thirds set has Fourier dimension 0.
The first examples of Salem sets of fractional dimension are due to Salem
[5] : Salem’s constructions were random fractals.
The well-approximable numbers were shown to be a Salem set by Kaufman,
[3]. This was later explained in detail by Bluhm, [2].
In the local-field setting, Papadimitropoulos modified Salem’s construction
to work over local fields. The goal of my joint project with Kyle is to show that
the well-approximable numbers in the discrete valuation ring Zp are a Salem
set.
Definition 0.1. The p-adic integers Zp are the set
ÿ
cj p j
j
where cj is a number from 0 to p ´ 1. Addition and multiplication of finite
sums is just like base-p arithmetic in Z. The topology is generated by the p-adic
absolute value, which is |x|p “ pj , where j is the first nonzero index in the sum.
With respect to this absolute value, Zp is a compact abelian group, and therefore
comes equipped with a finite Haar measure: the Haar measure of a ball is equal
to its radius.
Definition 0.2. The set of well-approximable numbers of degree τ in Zp is the
set
tx : there are infinitely many pairs r, q P Z such that |qx´r|p ď maxp|r|, |q|q´τ u.
It is easily seen from the Dirichlet principle that every x is approximable
of degree 2. A result of Morotskaya, together with a later result of Bernik and
Morotskaya, imply that the numbers approximable of degree τ has Hausdorff
dimension τ2 for any τ ą 2 [4] [1]. So the goal is to show that the Fourier
dimension of this set is also τ2 .
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I will now discuss what the additive characters on the p-adic numbers Qp
look like. There is a locally constant additive character χ such that χp1q “ ep p1 q
and χ is constant on cosets of pZp - note that it |x|p ă 1, then χpxq “ 1. Letting
χ be this character, each character of Qp is given by χpyxq for some y P Qp .
If |y|p ă 1, this character is trivial on Zp . So the family of characters can be
viewed as eptyxp´1 up q, where t¨up is the p-adic fractional part, and y is thought
of as an element of Qp {Zp .
Define Qpαq to be the set of p-adic integers of the form r{q where |r|, |q| ď pα
and pq, pq “ 1. Define Qpα,τ q to be the set of p-adic integers in the p´τ α thickening of this set. Define the function Fα to be the measure on Qpα,τ q
weighted by the number of pairs pr, qq corresponding to a given rational number.
ś
I will pick a quickly growing sequence tαj u and define µk “ kj“1 Fαj . I will
show that the µ
pk ’s are convergent in the supremum norm, so the µk ’s have a
nonzero weak limit µ, which will be our desired probability measure.
I will start by estimating the Fpα ’s. Observe that we can obtain µ
pk has a
k-fold convolution of the Fα ’s for appropriate α.
Note that Fα is constant on balls of radius p´τ α . This implies that Fpα psq “ 0
for any s with |s|p ą pτ α .
Now, it’s time to compute the Fourier coefficient Fpα psq for s such that |s|p ď
τα
p . If s is nonzero, we can write s “ s1 p´γ´1 , where s1 is a p-adic integer with
absolute value 1.
Fpα psq “
ż
Zp
“C
Fα psqeptsxuq dx
ÿ ż
r{qPQα
|x´r{q|p
epts1 x{pγ uq dx.
ăp´τ α
Here, C is an appropriate constant so that Fpα p0q “ 1. Now, the expression
in the exponent does not depend on anything other than the first γ digits of
s1 x (and therefore, not on any component of x other than the first γ digits).
Therefore, the integrand does not depend on x: because γ ă τ α, The integrand
is constant and equal to epts1 rq ˚ {pγ up q, where q ˚ denotes the inverse of q mod
pγ . So we are left with
ÿ
“ Cpτ α
eps1 rq ˚ {pγ q.
r{qPQα
We therefore only need to check the region where α ă γ ď τ α. We split this
sum up over different values of q:
ÿ ÿ
Cpτ α
eps1 rq ˚ {pγ q
qăpα rďpα
pq,pq“1
If we split this sum up over q, it’s trivial to see that this sum cancels if γ ď α.
`
˘´1
Now, we can use the
Evidently, the appropriate constant C is p´τ α #Qpαq
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geometric series bound and the trivial bound on the inner sum: the inner sum
is obviously less than or equal to pα in absolute value, but we also have that it’s
less than or equal to pγ {ps1 q ˚ q by the geometric series bound. So the goal is to
show that s1 q ˚ is usually large.
The trick here is simple: suppose s1 q ˚ ” y mod pγ , where y ă N and
q ă N :“ pα Then we have that s1 ” yq mod pγ . In other words, we have that
s1 ` zpγ “ yq for some z. But the right hand side is bounded above by M N , so
the left side is too, and there are only MN
pγ possible values of z. By the divisor
bound, each of these contributes no more than pM N qop1q possible solutions, so
we have a total of no more than
pMN q1`op1q
pγ
γ
such pairs q, y. This shows that s1 q ˚
α
1`op1q
times.
can only be smaller than N mod p at most pp Npqγ
Note that the statement that |s1 q ˚ | is no more than N mod pγ is equivalent
to saying that the angle ps1 q ˚ {pγ q is no larger than pNγ . So if we start with
N “ pγ´α , that will tell us the number of angles that are smaller than p´α .
This is the smallest angle we’re interested in because the trivial bound is better
for smaller angles.
So we apply the bound for each N from pγ´α to pγ . We then perform a dyadic
summation: The number of q such that s1 q ˚ is between 2j pγ´α and 2j`1 pγ´α is
no more than 2j - so each family of angles contributes approximately the same
amount to the sum- a total of pα`opαq . Because there are popαq summands, we
get the bound of pα`opαq on the sum.
Summarizing, we have the following bounds:
$
“ 1 if s “ 0
’
’
’
&“ 0 if 0 ă |s| ď pα
p
Fα psq
α
α
’
ď
p
if
p
ă
|s|p ď pτ α
’
’
%
“ 0 if |s|p ą pτ α
Let αj “ 2j . Then for large enough j, we have that for αj ă |s|p ď τ αj ,
µ
pk psq “ pFα1 ¨ ¨ ¨ Fαk q^ psq
“ ˚kℓ“1 Fppαℓ q psq
ÿ
Fpα1 ps1 q ¨ ¨ ¨ Fpαk psk q.
“
s1 `...`sk “s
Note that if sℓ is nonzero and in the support of Fpαpℓq for some ℓ ą j, the growth
rate of the αk ’s guarantees that we do not have s1 ` . . . ` sk “ s. Therefore,
this convolution operation stabilizes after j convolutions. Furthermore, it’s
immediately clear that we can’t have s1 ` . . . ` sj “ s if the sj term is zerothis would violate the ultrametric inequality. Therefore, we get that µ
pk psq “ 0
for k ă j and µ
pk psq “ µ
pj psq for k ě j.
We now need to find a bound on the number of decompositions for which
each piece s1 ` . . . ` sj “ s, where each |sj | ď pαj . Clearly, the digits of sj
after the p´αj´1 digit are uniquely determined. There are at most pαj´1 choices
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for the remaining digits. After making this choice, there are at most pαj´2
choices for sj´1 . Continuing in this manner, we get that there are a total of no
more than pαj´1 `¨¨¨`α1 terms in the sum, each of which has size no more than
p´αj `opαj q . Therefore, by choosing the αj to grow sufficiently rapidly, we can
guarantee that α1 ` . . . ` αj´1 À αj , so we get that the µ
pj psq À p´αj `opαj q .
In particular, this proves that the µ
pj ’s converge uniformly, because sups‰0 µ
pj psq Ñ
0 as j Ñ 8. Let µ
p be the limit of the µj ’s. We then know that the µj ’s converge
weakly to a measure µ with µ
p as its Fourier transform. Because of the bounds
´ 1 `op1q
p
on Fpαj q psq for each j, we have the desired bound: µ
ppsq ď |s|p τ
.
References
[1] V. I. Bernik and I. L. Morotskaya. Diophantine approximations in Qp and
Hausdorff dimension. Vestsı̄ Akad. Navuk BSSR Ser. Fı̄z.-Mat. Navuk, (3):3–
9, 123, 1986.
[2] Christian Bluhm. On a theorem of Kaufman: Cantor-type construction of
linear fractal Salem sets. Ark. Mat., 36(2):307–316, 1998.
[3] R. Kaufman. On the theorem of Jarnı́k and Besicovitch. Acta Arith.,
39(3):265–267, 1981.
[4] I. L. Morotskaya. Hausdorff dimension and Diophantine approximations in
Qp . Dokl. Akad. Nauk BSSR, 31(7):597–600, 668, 1987.
[5] R. Salem. On singular monotonic functions whose spectrum has a given
Hausdorff dimension. Ark. Mat., 1:353–365, 1951.
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