
TL;DR
This paper proves a new lower bound on the sum and product sets of certain discretized sets in [1, 2], improving previous results for larger values of .
Contribution
It introduces an improved bound on sum and product sets for -sets, advancing the understanding of sum-product phenomena in discretized settings.
Findings
Established a lower bound |A+A|+|AA| ^{-c}|A| for in (1/2, 1)
Improved upon previous bounds by Guth, Katz, and Zahl for large
Demonstrated the effectiveness of discretized sum-product estimates in additive combinatorics.
Abstract
Let be a -set with measure in the sense of Katz and Tao. For we show that for . This improves the bound of Guth, Katz, and Zahl for large .
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Discretized sum-product for large sets
Changhao Chen
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
Abstract.
Let be a -set with measure in the sense of Katz and Tao. For we show that
[TABLE]
for . This improves the bound of Guth, Katz, and Zahl for large .
Key words and phrases:
-sets, sum set estimates, Fourier transform
2010 Mathematics Subject Classification:
05B99
1. Introductions
Erdős-Volkmann [4] showed that for any there exists a subgroup of reals with Hausdorff dimension , and they conjectured that this property does not hold for subring of reals. Precisely the Erdős-Volkmann ring conjecture claims that there does not exists a subring of reals with Hausdorff dimension strictly between zero and one. Edgar and Miller [3] first proved this ring conjecture via the orthogonal projections of fractal sets. A slightly later Bourgain [1] independently proved the Erdős-Volkmann ring conjecture via the discretized ring conjecture (discretized sum-product) of Katz and Tao [10].
Discretized sum-product also has many other applications. For instance it is closely related to Falconer distance sets problem and the dimension of Furstenburg sets, see Katz and Tao [10] for more details. Bourgain [2] showed a different approach for the discretized sum-product, and given applications in the projections of fractal sets and Fourier analysis. For the applications of the discretized sets to projections of fractal sets see He [9], Orponen [12] and reference therein. For the applications of discretized sum-product to the Fourier decay of measures see Li [11].
We note that Bourgain [1, 2] does not produce an explicit bound. Recently Guth, Katz, and Zahl [7] given a short proof of the discretized sum-product theorem, and they showed an explicit bound for the discretized sum-product theorem. They [7] used some ideas from Garaev [6] and applied some discretized version of arguments from additive combinatorics.
We show some notation of Katz and Tao [10] first. Let be small and positive parameters. We use to denote , if , and if and . We say that a subset is called -discretized if is the union of intervals of lengths . For a positive constant we say that a subset is a -set if it is -discretized, and one has
[TABLE]
for all and .
We note that a -set here may have much smaller measure than . However we ask that any -set has positive measure to avoid to be an empty set.
We remark that the -set can be considered as the discrete approximation of set in at scale . Suppose that is a compact subset of . For each let
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and be the union of the interval which intersects , that is
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In many cases, the set is a -set for some . Here the parameter is often related to the “fractal dimension” of . For instance if is the classical Cantor ternary set, then for any the set is a -set. Indeed this follows from Falconer [5, Chapter 3], which shows that the box dimension of the Cantor set is . We remark that there are various dimensions in fractal geometry, we refer to [5] for more details.
The above argument shows that the -set in is also important for understanding the structure of set in . However in this project we consider -set on the line only.
Let . The sum sets of and is defined as
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and similarly the product sets of and is defined as
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Using the above notation, Bourgain’s discretized sum product theorem claims that for a -set with and the measure , there exists a constant such that
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Under the same condition, Guth, Katz, and Zahl [7] proved that for any one has
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By adapting the arguments of Garaev [6], Guth, Katz, and Zahl [7], and the bilinear bound of Bourgain [2, Theorem 7], we obtain the following.
Theorem 1.1**.**
Let and be a -set with measure . Then
[TABLE]
Note that for any , Theorem 1.1 gives a non-trivial lower bound. Furthermore for , Theorem 1.1 improves the bound of Guth, Katz, and Zahl [7].
2. Preliminaries
We use to denote the cardinality of set . Let be finite sets with , then the Ruzsa triangle inequality claims that
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See [13, Chapter 2] for a proof and many other useful sum sets estimates.
We need the following well known discretized version of Ruzsa triangle inequality. Our proof is based on Guth, Katz, and Zahl [7, Proof of Corollary 2.3], Orponen [12, Remark 4.40]. For many other discretized version of sum sets estimates see He [8], Tao [14].
We show a geometric observation first. Let be the union of disjoint intervals with length . Then for all we have
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Here we can change the parameter to any other fixed positive constant.
Lemma 2.1**.**
Let be -discretized sets. Then
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Proof.
With out losing general we may assume that each interval of and has length at least . In the end we change the estimate to , and this does not change our result.
For any set let . For any there exists such that
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It follows that
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Combining with (1) we obtain
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Applying Ruzsa triangle inequality to sets and applying (2) to and we obtain the result. ∎
Lemma 2.2**.**
Let be a -discretized set, and the union of the intervals of are pairwise disjoint. Then for any and any we have
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Proof.
For any and we have
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and hence
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Combining with (1) we finish the proof. ∎
Let be a function. The Fourier transform of the function at is defined as
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where throughout the paper we denote . Let be a measure on . The Fourier transform of the measure at is defined as
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For a subset we will also use to denote the characteristic function of . Let be two bounded sets. The convolution of and is defined as
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The (additive) energy of is defined as
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The second equality holds by applying the Plancherel identity and convolution theorem. Clearly we have
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By Cauchy-Schwarz inequality we obtain
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We will frequently use the Plancherel identity for a set. Precisely for a bounded subset we have
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We formulate the following version of Bourgain [2, Theorem 7]. Note that the interval is not essential, in fact Lemma 2.3 holds for any bounded interval.
Lemma 2.3**.**
Let be probability measures on such that for all and all ,
[TABLE]
Then for we have
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We remark that the statement of [2, Theorem 7] gives a bound for
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However the proof of [2, Theorem 7] (see [2, (8.3)]) indeed works for the bound
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which is used for bounding (22) below. Moreover the term appears, since the using of Cauchy-Schwarz inequality in the proof of [2, Theorem 7].
In particularly we have the following version for -sets which is easier for our using.
Lemma 2.4**.**
Let . Let and be -set and -set respectively. Then for any we have
[TABLE]
Proof.
For we define a measure by letting
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By the condition that is a -set we obtain
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Similarly for we define a measure by letting
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and we have
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Clearly we have
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Then Lemma 2.4 and estimates (6), (7) give the result. ∎
3. Proof of Theorem 1.1
By adapting the arguments in Guth, Katz, and Zahl [7] and especially in Garaev [6] we obtain the following.
Lemma 3.1**.**
Let . Let be a -set with measure . Then there exists a -set with measure
[TABLE]
such that for any we have
[TABLE]
Proof.
Let be a maximal -separated subset of , i.e., any two distinct elements of has distance at least , furthermore for any there exists such that . Note that
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and for all and ,
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Let
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Then we have
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and
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By Cauchy-Schwarz inequality we arrive
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Observe that
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Thus together with (10), (11), and (12), we obtain
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Thus there exists such that
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Let
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Then
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Taking dyadic decomposition for with , we obtain
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where
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and is an integer parameter such that
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Since the product set is a subset of , we have . It follows that
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Thus there exist and such that
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and for any we have
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Since and , we obtain
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and
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Now we intend to bound the measure of the set for each . For this purpose we introduce some notation first.
For each let For each let
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Note that
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and the intersection is a -discretized set for . Moreover for each by (13), (14) we have
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For any applying Ruzsa triangle inequality Lemma 2.1 for the sets and , we derive
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By (16) we have
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Since and the simper fact (1) we obtain
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Similarly, we have
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Combining with (17) and (18) we arrive
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Applying Lemma 2.2 we obtain that for any and we have
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Let
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We ask that is a small positive parameter, and hence . Furthermore, the estimates (8), (15) imply
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By (9) we obtain that is a -set which finishes the proof. ∎
Applying Lemma 2.4 we obtain the following upper bound of the mean value of energies .
Lemma 3.2**.**
Fix . Let be a -set with the measure . Let be a -set with measure . Then
[TABLE]
Proof.
For each and we have
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Thus by (3) and the condition we conclude that
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Let be a parameter which will be determined later. We decompose into three parts, and then bound (19) by three corresponding parts. Precisely,
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where
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[TABLE]
and
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For , we use the trivial bound , and we obtain
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For , clearly the trivial bound gives
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Applying Lemma 2.4, and the condition , we obtain
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Thus we arrive
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Combining with Fubini’s theorem and the condition , we have
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Plancherel identity (5) implies
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Thus we arrive
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Now we optimize the choice of the parameter to find the smallest upper bound for the parts . Recalling that we ask . In the end, the parameter , which makes the right hand sides of (21), (23) “comparable”, satisfies our need. Thus we derive
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Indeed the conditions , , and , imply that . It follows that
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Now we turn to the estimate for . By changing variables and applying the Plancherel identity we obtain
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Again by applying the Plancherel identity we have
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Combining with (20), (21), (24), (25), we obtain the desired bound. ∎
Now we turn to the proof of Theorem 1.1. Suppose that
[TABLE]
By Lemma 3.1 there exists a -set such that
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and for each we have
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Applying Lemma 3.2 to and , we conclude that there exists a such that
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By (4) and estimates (26), (27), we obtain that
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It follows that
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Note that for we have
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and hence
[TABLE]
which gives the result.
Acknowledgement
The author is grateful to Igor Shparlinski for his comments on the initial draft. Especially the author would like to thank the anonymous referee for carefully reading the manuscript and giving excellent comments, and thus improving the quality of this article.
This work was supported in part by ARC Grant DP170100786.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] J. Bourgain. On the Erdős-Volkmann and Katz-Tao ring conjectures. Geom. Funct. Anal 15(1): 334–365. 2003.
- 2[2] J. Bourgain. The discretized sum-product and projection theorems. J. Anal. Math. 112(1): 193–236. 2010.
- 3[3] G.A. Edgar and C. Miller. Borel subrings of the reals, Proc. Amer. Math. Soc. 131:4 (2003), 1121–1129.
- 4[4] P. Erdős and B. Volkmann. Additive Gruppen mit vorgegebener Hausdorffscher Dimension, J. Reine Angew. Math 221 (1966), 203–208.
- 5[5] K. J. Falconer, Fractal geometry: Mathematical foundations and applications , John Wiley, NJ, 2nd Ed., 2003.
- 6[6] M. Z. Garaev. An explicit sum-product estimate in 𝔽 p subscript 𝔽 𝑝 \mathbb{F}_{p} . Int. Math. Res. Notices. rnm 035. 2007.
- 7[7] L. Guth, N. Katz and J. Zahl. On the discretized sum-product problem. arxiv.org/abs/1804.02475
- 8[8] W. He. Sums, products and projections of discretized sets, Ph.D. thesis.
