A Non-Linear Roth Theorem for Fractals of Sufficiently Large Dimension
aa r X i v : . [ m a t h . C A ] M a y A NON-LINEAR ROTH THEOREM FOR FRACTALS OFSUFFICIENTLY LARGE DIMENSION
BEN KRAUSE
Abstract.
Suppose that d ≥
2, and that A ⊂ [0 ,
1] has sufficiently large dimension,1 − ǫ d < dim H ( A ) <
1. Then for any polynomial P of degree d with no constantterm, there exists { x, x − t, x − P ( t ) } ⊂ A with t ≈ P Introduction
In [3], the authors exhibit the existence of polynomial configurations in fractalsets; a key assumption on these fractal sets is that they have sufficiently large Fourierdimension, where we recall that the Fourier dimension of a set is given bydim F ( E ) :=sup { β : E supports a probability measure , µ, so that | ˆ µ ( ξ ) | ≤ C (1 + | ξ | ) − β/ } . The purpose of this short note is to show that – in one dimension – this phenomenonis independent on Fourier dimension of fractal sets, provided that E has sufficientlylarge Hausdorff dimension.In particular, we have the following result. Theorem 1.1.
Suppose that A ⊂ [0 , has sufficiently large dimension, − ǫ d < dim H ( A ) < , d ≥ . Then for any polynomial P of degree d with no constant term,there exists { x, x − t, x − P ( t ) } ⊂ A with t ≈ P . Acknowledgement.
The author would like to thank Alex Iosevich for his per-spective on the interplay between Fourier and Hausdorff dimension in detecting pointconfigurations.1.2.
Notation.
Here and throughout, e ( t ) := e πit . For real numbers A (typicallytaken to be dyadics), define f A to be the smooth Fourier restriction of f to | ξ | ≈ A ,similarly define f ≤ A , etc.For bump functions φ , we let φ j ( x ) := 2 j φ (2 j x ) . We will make use of the modified Vinogradov notation. We use X . Y , or Y & X ,to denote the estimate X ≤ CY for an absolute constant C . We use X ≈ Y asshorthand for Y . X . Y . We also make use of big-O notation: we let O ( Y ) denotea quantity that is . Y . If we need C to depend on a parameter, we shall indicate Date : May 21, 2019. this by subscripts, thus for instance X . p Y denotes the estimate X ≤ C p Y for some C p depending on p . We analogously define O p ( Y ).2. The Argument
Preliminaries.
We need the following Lemmas; the first is essentially a conse-quence of the main result [1], or [2].
Lemma 2.1.
Suppose that ≤ f i = φ t i ∗ f for some bounded function ≤ f ≤ [0 , , < t i < ∞ so that R f ≥ ǫ . Then there exists δ Roth ( ǫ ) > so that Z Z f ( x ) f ( x − t ) f ( x − P ( t )) dxdt ≥ δ Roth ( ǫ ) . The following refinement of [1, Lemma 5], due to [2], is our primary tool.
Lemma 2.2 (Lemma 1.4 of [2]) . Suppose that ˆ f is supported on | ξ | ≈ N . Then thereexists δ > so that (cid:13)(cid:13)(cid:13)(cid:13)Z f ( x − t ) f ( x − P ( t )) ρ ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) . N − δ · k f k k f k ; here ρ = ρ P is a C ∞ function, supported in an annulus {| t | ≈ P } , with derivative k ∂ l ρ k ∞ . P for all l ≤ L sufficiently large. Via the Fourier localization argument below, we see that this lemma essentiallyimplies a Sobolev estimate. Since sets of dimension > d E support a probabilitymeasure µ with k φ n ∗ µ k ∞ . n (1 − d E − κ ) , for some κ >
0, we see that µ is in the (negative) Sobolev class µ ∈ H κ ′ − (1 − d E ) / for any κ > κ ′ >
0; this is essentially the key to the argument.With these lemmas in hand, we turn to the proof.2.2.
The Proof of Theorem 1.1.
Proof.
Set f = f J := µ ∗ φ J for some sufficiently large J ; it suffices to exhibit upperand lower bounds on(2.3) Z Z f ( x ) f ( x − t ) f ( x − P ( t )) ρ ( t ) dtdx independent of J , for ρ an appropriate bump function. Split (2.3) into two terms:(2.4) Z Z f ( x ) f ( x − t ) f ≤ B ( x − P ( t )) ρ ( t ) dtdx ON-LINEAR ROTH 3 and its complementary piece(2.5)
Z Z f ( x ) f ( x − t ) f >B ( x − P ( t )) ρ ( t ) dtdx where B is a large parameter to be determined later. We begin with (2.4), which we write as
Z Z ˆ f ( − ξ − η ) ˆ f ( ξ ) d f ≤ B ( η ) m ( ξ, η ) dξdη, where m ( ξ, η ) := Z e ( − ξt − ηP ( t )) ρ ( t ) dt = m ( ξ, η ) · | ξ | . P | η | + | ξ |≫ P | η | · O B ((1 + | ξ | ) − B )=: m ( ξ, η ) + m ( ξ, η )by non-stationary phase considerations.Now, with B & A = A ( B, P ) ≫ B a large threshold, decompose (2.4) as a sum ofthree terms: (2.4) = Z Z f . A ( x ) f ≤ A ( x − t ) f ≤ B ( x − P ( t )) ρ ( t ) dtdx (2.6) + Z Z f ≫ A ( x ) f ≤ A ( x − t ) f ≤ B ( x − P ( t )) ρ ( t ) dtdx (2.7) Z Z f ( x ) f >A ( x − t ) f ≤ B ( x − P ( t )) ρ ( t ) dtdx. (2.8)The first term is a main term; an upper bound is simply given by(2.6) . k f . A k ∞ · k f ≤ A k ∞ · k f ≤ B k ∞ . A − d E ) ;as we will see, (2.7) and (2.8) are lower order error terms, so this upper boundmajorizes (2.4).As for the lower bound, an application of Lemma 2.1 yields a lower bound of(2.6) & δ Roth ( A d E − );the loss of A d E − comes from reproducing: we have k f . A k ∞ . A − d E . Since δ Roth ( ǫ ) grows super-polynomially in ǫ − , see [4], we stipulate that(2.9) 1 − C log B < d E < . In particular, we will choose B so large that for c ≥ c > δ Roth ( c ) ≫ B − δ , where δ is as in Lemma 2.2. BEN KRAUSE
The second term, (2.7), vanishes identically, since d f ≫ A ( − ξ − η ) d f ≤ A ( ξ ) d f ≤ B ( η ) = 0 . We express the third term using the Fourier transform:(2.8) =
Z Z ˆ f ( − ξ − η ) d f >A ( ξ ) d f ≤ B ( η ) m ( ξ, η ) dξdη + Z Z ˆ f ( − ξ − η ) d f >A ( ξ ) d f ≤ B ( η ) m ( ξ, η ) dξdη. The first term vanishes identically since | ξ | is so much larger that | η | , for an appro-priate choice of A . As for the error term, we estimate: (cid:12)(cid:12)(cid:12)(cid:12)Z Z ˆ f ( − ξ − η ) d f >A ( ξ ) d f ≤ B ( η ) m ( ξ, η ) dξdη (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Z | d f >A ( ξ ) | · | d f ≤ B ( η ) | · | m ( ξ, η ) | dξdη ≤ X N>A
Z Z | c f N ( ξ ) | · | d f ≤ B ( η ) | · | ξ | − B dξdη . B − d E / · A − d E / − B . A − d E − B =: A − C ;note the use of the trivial estimate k ˆ f k ∞ ≤ k f k in passing to the second line.In particular, taking into account (2.9), we have exhibited(2.10) (2.4) & δ Roth ( A d E − ) − A − C & δ Roth ( c )for some c ≥ c > A sufficiently large.We next term to (2.5), which we decompose as a sum of N > B :(2.4) =
Z Z f ( x ) f ( x − t ) f >B ( x − P ( t )) ρ ( t ) dtdx = X N>B
Z Z d f . N ( − ξ − η ) d f . N ( ξ ) c f N ( η ) m ( ξ, η ) dξdη (2.11) + X N>B
Z Z d f ≫ N ( − ξ − η ) d f ≫ N ( ξ ) c f N ( η ) m ( ξ, η ) dξdη (2.12)We begin with (2.12); by arguing as previously, the N th term admits an upper boundof N − C for a very large C = C ( B ), which leads to the estimate | (2.12) | . B − C . ON-LINEAR ROTH 5
It remains to consider (2.11); we extract the N th term once again, (cid:12)(cid:12)(cid:12)(cid:12)Z Z d f . N ( − ξ − η ) d f . N ( ξ ) c f N ( η ) m ( ξ, η ) dξdη (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Z d f . N ( − ξ − η ) d f . N ( ξ ) c f N ( η ) m ( ξ, η ) dξdη (cid:12)(cid:12)(cid:12)(cid:12) ≤ k f . N k ∞ · k Z f . N ( x − t ) f N ( x − P ( t )) ρ ( t ) dt k L . N − d E · N − δ · k f . N k . N − d E − δ . In passing from the first line to the second, we have (possibly) discarded O (1) termsof the form Z Z | [ f ≈ CN ( − ξ − η ) | · | \ f ≈ C ′ N ( ξ ) | · | c f N ( η ) | · | m ( ξ, η ) | dξdη for some large 1 ≪ C, C ′ . m ; but, on this domain, we retain the pointwise bound | m ( ξ, η ) | . (1 + | ξ | ) − B , sowe may handle these error terms as above.In particular, since (2.9) ensures that we have 1 − δ / < d E for sufficiently large B , we have exhibited a upper bound(2.13) (2.5) . B − δ − d E +2 . B − δ Combining (2.10) and (2.13), we see that we may estimate from below(2.3) ≥ δ Roth ( c ) − CB − δ , which yields the result. (cid:3) References [1] J. Bourgain. A nonlinear version of Roth’s theorem for sets of positive density in the real line.J. Analyse Math. 50 (1988), 169181. pages 2[2] P. Durcik, S. Guo, J. Roos. A polynomial Roth theorem on the real line, To appear in Trans.Amer. Math. Soc., arXiv:1704.01546. pages 2[3] K. Henriot, I. Laba, M. Pramanik. On polynomial configurations in fractal sets. Anal. PDE 9(2016), no. 5, 11531184. pages 1[4] R. Salem. Oeuvres Mathematiques, Hermann, Paris, 1967. pages 3
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