LARGE SUBSETS OF LOCAL FIELDS NOT CONTAINING
CONFIGURATIONS
ROBERT FRASER
Abstract. For certain families of functions {fq } mapping K nvq → K m ,
where K is a complete, nonarchimedean local field, we find a set E of large
Hausdorff dimension with the property that fq (x1 , . . . , xvq ) is nonzero for any
distinct points x1 , . . . , xvq ∈ E. In particular, this result can be applied to
show that the ring of integers of any local field contains a subset of Hausdorff
dimension 1 not containing any nondegenerate 3-term arithmetic progressions.
1. Introduction
We are interested in questions of the following form: given a family of functions
{fq } : Rnvq → K m , where K is a complete, nonarchimedean local field with ring
of integers R, how large can the Hausdorff dimension of the set E ⊂ Rn be if
E has the property that E does not contain any vq -tuples (x1 , . . . , xvq ) such that
fq (x1 , . . . , xvq ) = 0 for any q? Versions of this question have been considered for
the real numbers by Keleti [3], [4], Maga [5], Máthé [6], and Pramanik and the
author [2]. The objective of this paper is to establish local field versions of the
results [6] and [2], which will establish results similar to those occurring in [3] and
[4] as a special case.
We now state the main theorems of this paper.
Theorem 1.1. Let {fq } : Rnvq → R be a countable family of nonzero polynomials
of degree at most d whose coefficients are elements of R of finite height. Then there
exists a set E ⊂ R of Hausdorff dimension nd and Minkowski dimension n such
that, for all q, the set E does not contain vq distinct points x1 , . . . , xvq such that
fq (x1 , . . . , xvq ) = 0.
This theorem can be applied to the function x1 − 2x2 + x3 to show the existence
of a subset of any local field of Hausdorff dimension 1 that does not contain any
nontrivial 3-term arithmetic progressions. This contrasts with a recent result of
Ellenberg and Gijswijt [1], which states that for finite dimensional vector spaces
over (say) Z/3Z of dimension N , there is a constant C < 3 such that any set of
at least C N elements contains a 3-term arithmetic progression. In fact, there is a
set of Hausdorff dimension 1 with no v-tuple of distinct points satisfying any linear
equation of the form a1 x1 + · · · + av xv = 0 where a1 , . . . , av ⊂ R have finite height.
This implies in particular that if the characteristic of the field K is equal to zero,
then there are no solutions to any equations of this form with integer coefficients.
Theorem 1.2. For any ball B ⊂ R and integer v ≥ 3, let fq : R → K be a
countable family of functions in v variables with the following properties:
2010 Mathematics Subject Classification. 28A80.
1
2
ROBERT FRASER
(a) There exists rq < ∞ such that fq is rq times strictly differentiable.
(b) For each q, some partial derivative of fq of order rq ≥ 1 does not vanish at any
point of B v .
1
and Minkowski
Then there exists a set E ⊆ B of Hausdorff dimension at least v−1
dimension 1 such that fq (x1 , . . . , xv ) is not equal to zero for any v-tuple of distinct
points x1 , . . . , xv ∈ E and any function fq .
Theorem 1.3. Fix a ball B ⊂ R and positive integers m, n, v such that v ≥ 3,
and m ≤ n(v − 1). Let fq : (Rn )v → K m be a countable family of twice strongly
differentiable functions with the following property: the first derivative Dfq has full
rank on the zero set of fq for every q on B nv .
m
Then there exists a set E ⊆ B n of Hausdorff dimension at least v−1
and Minkowski
dimension n such that fq (x1 , . . . , xv ) is not equal to zero for any v-tuple of distinct
points x1 , . . . , xv ∈ E n and any function fq .
Theorem 1.4. Given any constant C > 0 and a vector α ∈ K v such that
(1)
α · u 6= 0 for every u ∈ {0, 1}v with u 6= 0, u 6= (1, 1, . . . , 1)
and such that
v
X
(2)
αj = 0,
j=1
there exists a positive constant c(α, K) and a set E = E(C, α, K) ⊆ [0, 1] of Hausdorff dimension c(K, α) > 0 with the following property.
The set E does not contain any nontrivial solution of the equation
f (x1 , · · · , xv ) = 0,
(x1 , · · · , xv ) not all identical,
for any twice strictly differentiable function f of the form
v
X
f (x1 , · · · , xv ) =
(3)
αj xj + G(x1 , · · · , xv )
j=1
(4)
where |G(x)| ≤ C
v
X
(xj − x1 )2 .
j=2
2. The arithmetic of complete, nonarchimedean local fields
Consider a complete discrete valuation ring R, the ring of integers of a field K.
Each element x of R can be written in the form
∞
X
xj tj
x=
j=0
where t is a uniformizing element of R and xj come from a set S of representatives
of the additive cosets of tR in R containing zero. The field R/tR is the residue
field of K and will be taken to have q = pf elements. Fixing such a set S, we will
call x0 , x1 , . . . the digits of x. We will refer to x0 as the least significant digit of
x, and the collection of digits x0 , x1 , . . . , xj−1 as the j least significant digits of x.
Say that x has height at most h with respect to x if xj = 0 for all j > h − 1; that
LARGE SUBSETS OF LOCAL FIELDS NOT CONTAINING CONFIGURATIONS
3
is, x has height at most h if all of the digits of x except possibly for the j least
significant digits are zero.
We will now prove the following basic arithmetic fact about local fields of characteristic zero:
Lemma 2.1. Let K be a complete nonarchimedean local field of characteristic zero
with ring of integers R. There exist a1 , . . . , ag depending only on the field K, such
that each element y of the set {−x : x ∈ R has height at most h} differs from aj in
only the h least significant digits for some j ≤ 1 ≤ d.
Proof. By the classification of local fields, K can be viewed as a finite extension of
Qp . The elements of Zp are of the form
X
x=
xe·j te·j
e·j≤h
where xj ∈ Fp , and e is the ramification index of the extension K/Qp . A simple
calculation in Qp shows that for such an x we have that −x is equal to
X
X
−x =
ye·j te·j +
(p − 1)te·j
e·j≤h
e·j>h
for some appropriate
P∞digits ye·j if x 6= 0, and 0 if x = 0. Notice that this expression
differs from −1 = j=0 (p − 1)te·j in only the first h digits xj .
Now, an arbitrary element of R can be written uniquely as
X
αk tj x
j≤e
k≤f
for some x ∈ Zp where αk form a vector space basis for the extension Fpf /Fp , where
f is the inertia degree of the extension. Evidently, multiplication by αk tj does not
change the validity of the argument before: therefore, it follows that −x can be
written as a sum of at most 2e·f summands of the form
X
X
ye·j+j0 te·j (αtj0 ) +
(p − 1)αte·j+j0 .
e·j<h
e·j≥h
Again, we have that the above expression differs from −αtj0 in only the first h
digits xj . So by the linear independence of the αk tj0 , it follows that the sum differs
from one of the sums
X
−ǫj0 ,k αk tj0
0≤j0 <e
0≤k<f
for ǫj0 ,k = 0 or 1, in only the first h digits.
The characteristic p version of the above result is even easier: the negative
of an element of finite height is another element of the same height, since every
complete, nonarchimedean local field of characteristic p is isomorphic to some finitedimensional unramified extension of Fp ((t)), the field of Formal Laurent series over
Fp .
The values a1 , . . . , ag from the arithmetic lemma all have digital expansions that
are eventually repeating, and since the tails of these numbers are the only thing
that matter for the statement of the lemma, we can take these numbers to have
repeating expansions. We will freely do so.
4
ROBERT FRASER
It will be convenient to know how far these numbers are from 0. Define the
quantity βK to be the maximum number of consecutive zeroes appearing in the
nonzero a1 , . . . , ag . It is not difficult to check that this quantity βK is simply equal
to e − 1, where e is the ramification index of the extension K/Qp , but this fact is
not used anywhere in this paper and will therefore not be shown. Then βk is useful
for the following reason: if δ is any nonzero element of R with absolute value at
most q −βK −h , and x and y are any elements of R of height at most h, then the
quantity −x + δ satisfies |(−x + δ) − y| ≥ δ, where the absolute value is chosen so
that |t| = q.
This is useful when combined with the following algebraic fact: If p is a v-variate
polynomial of degree d with coefficients in R of height at most b, and x1 , . . . , xv are
elements of R with height at most ν, then we know that p(x1 , . . . , xv ) is a sum of
at most s terms of height at most b + νd, or negatives thereof, where s := d+v
is
v
the dimension of the space of polynomials of degree at most d in v variables. We
can then rewrite this quantity as a sum of two terms, one of which is an element of
K of height at most b + νd + s, and the other of which is the negative of an element
of K of height at most b + νd + s.
Thus, we have the useful fact that p(x1 , . . . , xv ) + δ, for p, x1 , . . . , xv as before
and |δ| < q −βK −b−νd−s ,is not within |δ| of zero.
This fact will serve as a local-field substitute for the following algebraic fact: if
p is a nv-variate polynomial of degree d with coefficients in Z, and x1 , . . . , xv are
multiples of N1 , then p(x1 , . . . , xv ) is a multiple of N1d , so p(x1 , . . . , xv ) + δ differs
from zero by at least |δ| so long as |δ| < 2N1 d . This simple algebraic fact is the key
to Máthé’s construction in [6].
The only remaining
of the puzzle is the familiar fact that if a polynomial
piece
∂p −C
p satisfies q
< ∂xv uniformly on a compact set T1 × · · · × Tv where, then
|q −C δ| ≤ |p(x1 , . . . , xv + δ) − p(x1 , . . . , xv )| ≤ δ for all δ sufficiently small (recalling
here that any polynomial with coefficients in R has derivatives bounded in absolute
value by 1 on R). This implies, in light of the above arguments, that if δ is smaller
than q −βK −b−νd−s , and x1 , . . . , xv all have height at most N , then p(x1 , . . . , xv + δ)
is not within q −C |δ| of 0.
Finally, since the partial derivatives of p are all bounded above by 1 in absolute
value, we have that the same bound holds provided that x1 , . . . , xv are all within
balls of radius q −βK −b−νd−s−C−1 around elements of R of height at most ν. This
establishes the following result:
Proposition 2.2. Let T1 , . . . , Tv−1 , Tv be sets, each of which is a union
of balls
∂p 1
of radius qµ , and let p(x1 , . . . , xv ) be a polynomial satisfying ∂xv ≥ q −C for
(x1 , . . . , xv ) ∈ T1 × · · · × Tv . Let ǫ > 0. Then there exists ν0 depending on ǫ, C,
and µ with the following property: Subdivide each such ball into balls of radius q1ν ,
where ν is any integer satisfying ν ≥ ν0 . Then there exist sets S1 ⊂ T1 , . . . , Sv ⊂ Tv
such that:
(a) There are no solutions to p(x) = 0 with x1 ∈ S1 , . . . , xv ∈ Sv . Furthermore, p
satisfies the bound |p(x1 , . . . , xv )| ≥ q −ν(d−ǫ) on S1 × · · · × Sv .
(b) For i = 1, . . . , v − 1 and any ball U of radius q1ν contained in one of the q1µ -balls
of Ti , we have that Si ∩ U is a ball of radius ≥ q −ν(d−ǫ) .
LARGE SUBSETS OF LOCAL FIELDS NOT CONTAINING CONFIGURATIONS
5
There is nothing special about xv in the statement of Proposition 2.2: it is simply
stated in terms of the xv variable for convenience, and we will frequently apply it
to variables other than the xv variable.
As a point of notation, we will frequently refer to q ν as N , q ν0 as N0 , and q µ as
M in section 4.
Proof. Select ν0 > M sufficiently large that q −βK −b−ν0 d−s−C−1 is smaller than
q −ν0 (d−ǫ) , and apply the above argument to all of the q −ν balls contained in T1 ×
· · · × Tv
3. Avoidance of Configurations at a Single Scale
The lemma proven at the end of the previous section will be central to the
construction described in Theorem 1.1. In order to prove Theorems 1.2 and 1.3, we
will need p-adic analogues of the lemmas appearing in [2]. The first step in proving
such an analogue is counting the balls intersected by the zero set of f .
Lemma 3.1. There exists M0 (C0 , C1 , C2 ) > 0 such that the following statement
1
and let
holds for all M ≥ M0 . Let T be an nv-dimensional ball of radius M
m
f (x1 , . . . , xv ) : T → R be a function such that Df has an m-by-m minor that
is, in absolute value, at least a constant C0 on all of T, and whose entries are
bounded above in absolute value by a constant C1 . Suppose further that C2 is an
upper bound for the operator norm of the second derivative of f . Let Zf be the set
of (x1 , . . . , xv ) with f (x1 , . . . , xv ) = 0. Subdivide the ball T into balls of side length
ℓ. If ℓ is sufficiently small, then the number of balls that are intersect the zero set
1 n−mv
of f is at most C3 M
ℓ
, where M0 and C3 depend on C0 , C1 , and C2 but not
on ℓ.
Proof. Let x0 be a point in Zf . Consider the vector space V spanned by the rows
of Dfx0 . Consider points of the form x0 + u where u is a vector in the vector space
1
. By the assumptions on Df , we have f (x0 + u) =
V of magnitude at most M
2
f (x0 ) + Dfx0 u + O(kuk ). Here, f (x0 ) = 0, and kDfx0 uk ≥ K0 kuk, where K0 is a
constant depending on C0 and C1 . So we have that kf (x0 + u)k ≥ C0 kuk−O(kuk2 ).
This is guaranteed to be positive provided that M is sufficiently large.
1
Let ℓ be smaller than M
. For a given x ∈ Zf , consider the slab consisting of
points x+V +w where w ∈ ker(Dfx0 ) satisfies |w| < ℓ. Let x+v+w be some point in
this slab. We have that f (x+ v + w) = f (x)+ Dfx (v + w)+ O(kv + wk2 ) = Dfx (v +
2
2
2
w) + O(kv + wk ). This has norm at least (K0 − C
M ) kvk − K1 kwk + O(kv + wk ),
where K1 is a constant depending only on C1 . f (x + v + w) is therefore nonzero
K1
provided that kvk is at least K0 −C
kwk. Recall that kwk < ℓ, so if M is
2 /M
2K1
ℓ. Thus,
K0 m
1
subdividing this slab into ℓ-balls, we have Zf will intersect only at most 2K
K0
1
balls in the slab. Taking the union over disjoint parallel slabs that cover the M
m
K1
ℓm−nv /M intersections.
ball, we have a total of O
K0
sufficiently large this inequality will hold provided that kvk is at least
6
ROBERT FRASER
This fact will allow us to apply the exact same combinatorial argument appearing
in [2] to arrive at the following proposition.
Proposition 3.2. Let T1 , . . . , Tv be disjoint sets that can be expressed as the union
1
of balls of radius M
. Let f (x1 , . . . , xv ) be a function defined on T1 × · · · × Tv with
the property that the derivative of f has full rank on T1 × · · · × Tv . Then there
exists an N0 such that for all N > N0 there exist S1 ⊂ T1 , . . . , Sv ⊂ Tv satisfying
the following conditions:
(1) There are no solutions to f (x1 , . . . , xv ) = 0 with x1 ∈ S1 , . . . , xv ∈ Sv .
(2) For any 1 ≤ j ≤ v − 1, and each ball B of radius N1 contained in Tj , we
c
for some constant
have that B ∩ Sj is a ball of radius approximately N v−1+ǫ
c.
1
(3) For each ball B of radius N1 contained in Tj , except for at most a M
fraction, we have that B ∩ Sv is a union of balls of radius approximately
c
N v−1+ǫ for some small constant c.
The proof of this proposition, given the lemma, is a purely combinatorial argument identical to the one appearing in [2].
4. The construction of the sets in Theorems 1.1, 1.2, and 1.3
The proofs of Theorems 1.1, 1.2, and 1.3 from the propositions above proceed
in almost the same way as in [2]. We will keep a running queue Q that will guide
the construction. As before, in the case of theorems 1.1 and 1.2, we choose a
sequence of “privileged” differential operators D1 , . . . , Drq , where Drq represents
a partial rq th derivative of f that is known to be nonvanishing on B nvq . Such
a partial derivative exists for Theorem 1.2 by assumption and for Theorem 1.1
because a nonzero polynomial always has some partial derivative that is constant
and nonzero. Let E0 be the ball B n .
One minor difference in the proof of Theorem 1.1 is that the number of variables
v is allowed to vary. For the cases of Theorems 1.2 and 1.3, take v = v1 = v2 = · · ·
throughout the rest of this argument.
We now select an M0 sufficiently large so that B n contains at least v1 balls
(0)
(0)
of radius M0 . Let B1 , . . . , BM0 be an enumeration of the balls of radius M0
contained in B and let σ0 be the family of v1 -tuples of distinct such balls, ordered
lexicographically (identified in the usual way with the family of functions from
{1, . . . , v1 } into {1, . . . , M0 }). Q0 be the queue consisting of 4-tuples:
{(1, k, σ, 0) : 0 ≤ k ≤ r1 − 1, σ ∈ Σ0 }
where the queue elements are ordered so that (1, k, σ, 0) precedes (1, k ′ , σ ′ , 0) whenever σ < σ ′ and (1, k, σ, 0) precedes (1, k ′ , σ, 0) whenever k > k ′ .
At stage 1, we apply the appropriate proposition (that is, Proposition 2.2 in
the case of Theorem 1.1 and Proposition 3.2 in the cases of Theorems 1.2 and
(0)
(0)
1.3) to the function f = Dk f1 on the sets T1 = Bσ(1) , . . . , Sv = Bσ(k) , and to a
large number N1 to arrive at the sets S1 , . . . , Sv . Define E1 ∩ Tj to be Sj , and
E1 ∩ (T1 ∪ · · · ∪ Tv )c = E0 ∩ (T1 ∪ · · · ∪ Tv )c . Let E1 denote the family of balls of
c
that make up E1 . Let L1 denote the number of balls in
radius approximately N v−1
E1 and let Σ1 denote the family of functions from {1, . . . , v} into {1, . . . , L1 }. We
(1)
(1)
will refer to the balls in E1 as B1 , . . . , BL1 . Let Q′1 denote the queue of 4-tuples:
{(q, k, σ, 1) : 1 ≤ q ≤ 2; 0 ≤ k ≤ rq − 1, σ ∈ Σ1 }.
LARGE SUBSETS OF LOCAL FIELDS NOT CONTAINING CONFIGURATIONS
7
where (q, k, σ, 1) precedes (q ′ , k ′ , σ ′ , 1) if σ < σ ′ ; (q, k, σ, 1) precedes (q ′ , k ′ , σ, 1) if
q < q ′ , and (q, k, σ, 1) precedes (q, k ′ , σ, 1) if k > k ′ . Append the queue Q′1 to the
queue Q0 to arrive at the queue Q1 .
At stage j, we consider the jth queue element: say (q, k, σ, j ′ ). The assumptions
on fq , together with the ordering of the queue, will imply (as in [2]) that Dk+1 fq
(j ′ )
(j ′ )
is nonzero whenever (x1 , . . . , xv ) is in the set Bσ(1) × · · · × Bσ(v) . Thus we can
apply the proposition. Pick Nj to be the largest of M0 (where M0 is as appears in
the proposition), eNj−1 , and vj . We apply the proposition to T1 = Bσ(1) , . . . , Tv =
Bσ(v) , f = Dk fq , N = Nj . Take Ej ∩ T1 = S1 , . . . , Ej ∩ Tv = Sv . Finally, take
Ej ∩ (T1 ∪ · · · ∪ Tv )c = Ej−1 ∩ (T1 ∪ . . . ∪ Tv )c . Enumerate the balls that constitute
(j)
(j)
Ej as B1 , . . . , BLj . Let Σj denote the family of functions from {1, . . . , v} to
{1, . . . , Lj }. Form the queue Q′j consisting of 4-tuples of the form
{(q, k, σ, j) : 1 ≤ q ≤ j, 0 ≤ k ≤ rq − 1; σ ∈ Σj }
and append this queue to the existing queue Qj−1 to form a new queue Qj .
We can see that, for any q, it follows by the same argument as in [2] that there
are no solutions to fq (x1 , . . . , xv ) = 0. Also, the Hausdorff dimension of the set
E = ∩j Ej is as desired, again by the same argument as in [2]. The argument for
the Minkowski dimension is slightly different.
Lemma 4.1. The Minkowski dimension of the set E is equal to 1 (in the case of
Theorem 1.2) or n (in the case of Theorems 1.1 and 1.3).
Proof. Let Nℓ (E) be the number of closed balls of radius ℓ that are required to
cover E. Notice that this is equivalent to the number of balls of radius ℓ that
intersect E. So we need to count the number of such balls that intersect E. Let ℓj
denote the radius of the balls in Ej .
If ℓ = ℓj for some j, and B is a ball of radius ℓj−1 that is not contained in
T1 , . . . , Tv at stage j of the construction,
then the number of balls of radius ℓj
ℓ
n
j−1
required to cover B is precisely
. This is ≥ ℓj−n+ǫ (provided that ℓ is
ℓj
sufficiently small) by the growth rate of the Nj .
Now suppose ℓj+1 < ℓ < ℓj , and consider any ball B of radius ℓj−1 such that B
is not contained in T1 ∪ · · · ∪ Tv at either stage j or stage j + 1 of the construction.
Such a ball will always exist provided that j is large enough.
Then by the argument from before, E intersects all of the balls of radius ℓ
contained in B, and thus at least ℓj−n+ǫ of radius ℓj inside the ball B. Evidently
n
ℓ
· ℓj−n+ǫ ≥ ℓ−n+ǫ of the balls of radius ℓ, as
E must then intersect at least ℓj
desired.
This completes the proofs of Theorems 1.1, 1.2, and 1.3.
5. Simultaneous avoidance of configurations with the same
linearization
The other result applies to (possibly uncountable) family of functions f (x1 , . . . , xv )
that have the property that they have agreeing, unchanging, nondegenerate linearizations along the diagonal. The specific conditions are outlined in the statement
of theorem 1.4. This is a p-adic analogue of a theorem appearing in [2].
8
ROBERT FRASER
As in the case of the Theorems 1.2, and 1.3, the construction follows from a
Cantor-like set in exactly the same manner as [2], so we will only prove the key
proposition for the construction. This proposition outlines the strategy for a single
step of the construction.
Lemma 5.1. Let B1 , B2 ⊂ B be distinct (and thus disjoint) balls of radius r. Let
A ⊂ {1, . . . , v} be an arbitrary nonempty proper subset of {1, . . . , v}. Then we can
find B1′ ⊂ B1 , B2′ ⊂ B2 of radius at least C1 r such that |α1 x1 + · · · + αv xv | ≥ C1 r
for xj ∈ B1′ if j ∈ A and xj ∈ B2′ if j ∈
/ A, where C1 is a constant depending on
the vector α and the local field K.
Proof. Since the bounds in this lemma are allowed to depend on α, we will normalize
α so that every element of α is in R and at least one component of α is invertible
in R; that is, the norm of α will be taken to be
1.
P
−1 Assuming this normalization, let C1 = q j ∈A
/ αj . This was assumed to be
strictly positive in the statement of Theorem 1.4.
Then, selecting B1′ and B2′′ to be any balls of radius C1 r contained in B1 and
B2 , we consider the set α · B, where B = B (1) × · · · × B (v) , and B (j) = B1′ if j ∈ A
and B (j) = B2′′ otherwise. Since each component of α is in R, we have immediately
that α · B is contained in a ball A ⊂ R of radius at most C1 r. If this ball is not the
unique C1 r-ball containing zero, then we can take B2′ = B2′′ to complete the proof.
If not, then select any element b ∈ R with |b| = q −1 r and define B2′ := B2′′ + b ⊂ B2 .
′
Then defining the ball B ′ to be B (1)∗ × · · · × B (v)∗ , with B (j)∗
P = B1 if j ∈ A and
(j)∗
′
′
B
= B2 otherwise, we have that α · B maps into A + b j∈AC αj , which is a
distinct ball of radius C1 r.
This lemma can be applied iteratively in the following way: Starting with a ball
B of radius r, we can pick two balls B1 , B2 ⊂ B of radius q −r . Then, for some
A, apply Lemma 5.1 to arrive B1′ and B2′ . Now, pick a different A and apply the
lemma to these balls to arrive at B1′′ and B2′′ . Repeat this process for all nonempty
proper subsets A ⊂ {1, . . . , v}, and we arrive at B1∗ ⊂ B1 , B2∗ ⊂ B2 where B1 and
B2 have radius at least C ∗ r for some C ∗ , and α1 x1 + · · · + αv xv is at least C ∗ r
in absolute value for all x1 , . . . , xv ∈ B1∗ ∪ B2∗ not all coming from B1∗ and not all
coming from B2∗ .
Proposition 5.2. There exists C ∗ depending only on α and on the local field K with
the following property. For any ball B with r := radius(B), there exist B1∗ , B2∗ ⊂ B
such that B1∗ , B2∗ are balls of radius C ∗ and such that α1 x1 + · · · + αv xv has absolute
value at least C ∗ r for x1 , · · · , xv ∈ B1∗ ∪ B2∗ provided that not all of the x1 , · · · , xv
are in B1∗ , and not all of the x1 , . . . , xv are in B2∗
The rest of the proof proceeds exactly as in [2]: Starting with a ball of small
enough radius for the error terms in approximating f (x1 , . . . , xv ) by α1 x1 + · · · +
αv xv to be inconsequential, we apply Proposition 5.2 on successively smaller scales
to arrive at a set of positive Hausdorff dimension.
References
[1] J. Ellenberg and D. Gijswijt. On large subsets of fqn with no three-term arithmetic progression.
Available from: https://arxiv.org/abs/1605.09223.
[2] R. Fraser and M. Pramanik. Large sets avoiding patterns. Available from: https://arxiv.
org/abs/1609.03105.
LARGE SUBSETS OF LOCAL FIELDS NOT CONTAINING CONFIGURATIONS
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[3] Tamás Keleti. A 1-dimensional subset of the reals that intersects each of its translates in at
most a single point. Real Anal. Exchange, 24(2):843–844, 1998/99.
[4] Tamás Keleti. Construction of one-dimensional subsets of the reals not containing similar
copies of given patterns. Anal. PDE, 1(1):29–33, 2008. Available from: http://dx.doi.org/
10.2140/apde.2008.1.29.
[5] Péter Maga. Full dimensional sets without given patterns. Real Anal. Exchange, 36(1):79–90,
2010/11. Available from: http://projecteuclid.org/euclid.rae/1300108086.
[6] András Máthé. Sets of large dimension not containing polynomial configurations, 2012. Available from: http://arxiv.org/pdf/1201.0548v1.pdf.
Department of Mathematics, University of British Columbia, Canada
E-mail address: [email protected]