Extension theorems for the Fourier transform associated with nondegenerate quadratic surfaces in vector spaces over finite fields

Iosevich, Alex; Koh, Doowon

doi:10.1215/ijm/1248355353

Cited by 17 publications

(36 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recall that if d = 2, then the exponent 4/3 gives a much better result than the exponent (d + 1)/2 which is optimal for odd dimensions. When d ≥ 3, the L 2 − L (2d+2)/(d−1) extension result for spheres is also known in [10] and can be also applied to the Erdős-Falconer distance problem but we can only obtain the exponent (d + 1)/2.…”

Section: Results On the Restriction Theorem For Spheresmentioning

confidence: 96%

“…The extension problem for spheres is more delicate than that of paraboloids, and it was studied by Iosevich and Koh. In [10], they obtained the sharp L 2 − L 4 extension result for circles, which the authors of [2] applied to deduce the exponent 4/3 for the Erdős-Falconer distance problem in dimension two. Recall that if d = 2, then the exponent 4/3 gives a much better result than the exponent (d + 1)/2 which is optimal for odd dimensions.…”

Section: Results On the Restriction Theorem For Spheresmentioning

confidence: 99%

See 1 more Smart Citation

On the Sums of Any $k$ Points in Finite Fields

Covert¹,

Koh²,

Pi³

2016

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Results On the Restriction Theorem For Spheresmentioning

confidence: 96%

Section: Results On the Restriction Theorem For Spheresmentioning

confidence: 99%

On the Sums of Any $k$ Points in Finite Fields

Covert¹,

Koh²,

Pi³

2016

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…We note that a smaller exponent gives a better extension result. The first result in Proposition 1.1 was given in [6] and it gives the Stein-Tomas result, the L 2 → L (2d+2)/(d−1) bound. It is well known in [7] that the Stein-Tomas result gives the optimal L 2 → L r estimate for spheres in general odd dimensions.…”

Section: Introductionmentioning

confidence: 98%

On Restriction Estimates for the Zero Radius Sphere over Finite Fields

Iosevich¹,

Koh²,

Pham³

et al. 2020

Can. J. Math.-J. Can. Math.

Self Cite

View full text Add to dashboard Cite

We study the finite field Fourier restriction/extension problem for spheres in even dimensions d ≥ 4. We prove that the L p → L 4 extension estimate for spheres of non-zero radii holds for 4d 3d−2 ≤ p ≤ ∞. Our result is sharp and improves the L (12d−8)/(9d−12)+ε → L 4 extension result for all ε > 0 due to the first and second listed authors [7]. The key new ingredient is improved additive energy estimates for subsets of spheres in even dimensions. In particular, our additive energy estimate improves and extends Rudnev's recent work [15] in four dimensions to higher even dimensions.As the most interesting result of this paper, we prove that if −1 is not a square number of F * q and the dimension d is 4k + 2 for some k ∈ N, then the L 2 → L (2d+4)/d extension estimate for spheres of zero radius holds. This result is also sharp and provides us of an interesting fact that the L 2 → L r extension estimate for zero spheres with specific assumptions is much better than the Stein-Tomas result which can not be improved in general for cones or zero spheres in even dimensions.2010 Mathematics Subject Classification. 52C10, 42B05, 11T23 .

show abstract

“…See, for example, [6,7,9]. Mockenhaupt and Tao [9] first posed the extension problems in the finite field setting for various algebraic varieties S and they obtained reasonably good results from extension problems for paraboloids in d-dimensional vector spaces over finite fields.…”

Section: Introductionmentioning

confidence: 99%

Extension theorems for paraboloids in the finite field setting

Iosevich

Koh

2009

Math. Z.

Self Cite

View full text Add to dashboard Cite

In this paper we study the L p − L r boundedness of the extension operators associated with paraboloids in F d q , where F q is a finite field of q elements. In even dimensions d ≥ 4, we estimate the number of additive quadruples in the subset E of the paraboloids, that is the number of quadruples (x, y, z, w) ∈ E 4 with x + y = z + w. As a result, in higher even dimensions, we obtain the sharp range of exponents p for which the extension operator is bounded, independently of q, from L p to L 4 in the case when −1 is a square number in F q . Using the sharp L p − L 4 result, we improve upon the range of exponents r , for which the L 2 − L r estimate holds, obtained by Mockenhaupt and Tao (Duke Math 121: 2004) in even dimensions d ≥ 4. In addition, assuming that −1 is not a square number in F q , we extend their work done in three dimension to specific odd dimensions d ≥ 7. The discrete Fourier analytic machinery and Gauss sum estimates make an important role in the proof.

show abstract

Extension theorems for the Fourier transform associated with nondegenerate quadratic surfaces in vector spaces over finite fields

Cited by 17 publications

References 10 publications

On the Sums of Any $k$ Points in Finite Fields

On the Sums of Any $k$ Points in Finite Fields

On Restriction Estimates for the Zero Radius Sphere over Finite Fields

Extension theorems for paraboloids in the finite field setting

Contact Info

Product

Resources

About