Restriction estimates of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-removal type for k-th powers and paraboloids

We obtain restriction estimates of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-removal type for the set of k-th powers of integers, and for discrete d-dimensional surfaces of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \{ (n_1,\dots ,n_d,n_1^k + \cdots + n_d^k) \,:\, |n_1|,\dots ,|n_d| \leqslant N \}, \end{aligned}$$\end{document}{(n1,⋯,nd,n1k+⋯+ndk):|n1|,⋯,|nd|⩽N},which we term ‘k-paraboloids’. For these surfaces, we obtain a satisfying range of exponents for large values of d, k. We also obtain estimates of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-removal type in the full supercritical range for k-th powers and for k-paraboloids of dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d < k(k-2)$$\end{document}d


Introduction
We are interested in restriction theorems for discrete surfaces in Z d . We restrict our attention to parametric surfaces of the form where P = (P 1 , . . . , P r ) is a system of r integer polynomials in d variables, and we assume that the map P : Z d → Z r is injective for simplicity. When the polynomials P 1 , . . . , P r have degree k 1 , . . . , k r , we define the total degree of the system P as K = k 1 + · · · + k r . We denote the action of the extension operator on a sequence a : The natural restriction conjecture, based on heuristics from the circle method, is that the ε-free estimate holds for 2 p < 2K d . This conjecture has to be corrected when the discrete surface {P(x), x ∈ Z d } contains large special subvarieties, but this does not appear to be the case for the surfaces we study.
In the supercritical range, Bourgain resolved the natural restriction conjecture in the case P = (x 2 ) of the squares [2] and in the case P = (x, x 2 ) of the 2D parabola [3], via discrete versions of the Tomas-Stein argument [25,Chapter 7] and the Hardy-Littlewood circle method. Bourgain and Demeter [4] later established the ε-full estimate (1.3) for arbitrary definite irrational paraboloids P = (x 1 , . . . , x d , θ 1 x 2 1 + · · · + θ d x 2 d ) with θ i ∈ (0, 1] in the full supercritical range p 2(d+2) d , by developing powerful methods of multilinear harmonic analysis (the indefinite case was later resolved in [5]). In the rational case θ 1 = · · · = θ d = 1, the ε-loss can be eliminated via Bourgain's earlier work [3]. In an important recent work, Killip and Vişan [16] removed the ε-loss for all definite parabolas, using new techniques partly inspired by Bourgain's [3]. This note relies only on the earlier number-theoretic approach of Bourgain [2,3], albeit with significant modifications, since it is more adapted to our objective. Indeed, we primarily seek to obtain weaker estimates of the form for a certain ζ > 0, in the complete supercritical range of exponents q > 2K d , or a good approximation thereof. We succeed in doing so for several classes of surfaces generalizing that of the squares and the parabola. Moreover, our precise values of ζ are obtained by making use of the recent resolution of the Vinogradov mean value conjectures in [26] and [6].
Before introducing these results, we discuss our motivation to seek ε-removal estimates of the form (1.5). Justifying their terminology, these estimates can be used to remove the extraneous factor N ε in (1.3), as recalled in Lemma 3.1 below. Methods of multilinear harmonic analysis [4], or even moment bounds exploiting arithmetic information typically produce a factor of this form. While the N ε factor is sometimes inconsequential, the sharp estimate (1.2) is often necessary in applications to additive combinatorics. More specifically, restriction estimates are of key importance in the study of linear equations of the form s i=1 λ i P(n i ) = 0, where the λ i are non-zero integer coefficients summing to zero and the variables n i lie in a sparse subset of {1, . . . , N } d (or in a sparse subset of (P ∩ {1, . . . , N }) d , where P are the prime numbers). When the system of polynomials P is translation-invariant 1 , this system of equations can be studied via density-increment-based strategies [10,11,15,20] exploiting L ∞ → L p or L 2 → L p restriction estimates for the surface (1.1); we refer to [11] for a more complete discussion. In the general case, one can also analyze such systems by transference-based strategies [7][8][9] which rely only on L ∞ → L p estimates, although these take a more complicated shape due to the presence of the W -trick.
Note also that truncated estimates of the form (1.5) can be completed into full estimates of the form (1.2) for a large enough range of exponents, whenever a subcritical estimate of the form (1.4) is known (which is always the case for p = 2). This familiar procedure is recalled in Lemma 3.6 below, but it generally gives a poor range of exponents due to the smallness of the parameter ζ , which is related to Weyl exponents.
The first surface we study is corresponding to the system of polynomials P = (x k ) of total degree k, when k 3 is an integer. In this case we obtain the complete supercritical range of exponents for (1.5) and an incomplete range of exponents for truncated restriction estimates of the form (1.2). Theorem 1.1 Let k 3 and τ = max(2 1−k , 1 k(k−1) ), and write P = (x k ). The estimate (1.5) holds for any p > 2k and ζ < τ/2, and the estimate (1.2) holds for p > 2 + 2(k − 1)/τ .
The proof of this result consists in an adaptation of Bourgain's argument for squares [2]. We comment in Sect. 5 on the results that can be obtained for arbitrary monomial curves by this approach. It turns out that one only obtains the whole supercritical range for the curve (n k ), due to the lack of efficient majorants of Weyl sums on major arcs in other cases.
Let R s,k (n) denote the number of representations of n as a sum of s k-th powers of integers. Hypothesis K of Hardy and Littlewood [24,Section 17] states that R k,k (n) ε n ε for k 2. It is known (and easy to show) for k = 2, and while it has been disproved for k = 3 by Mahler [17], it remains open for k 4. Under this strong hypothesis, which is far out of reach of current methods, our epsilon-removal estimate implies the full supercritical range of conjectured restriction estimates for k-th powers.

Corollary 1.2 Let k 3 and write
Fix a dimension d 1 and a degree k 3. The next surface we study is the which is the usual paraboloid when k = 2. Our first theorem simplifies the approach of Bourgain for the parabola [3]; the cost of our simplification is that we do not acquire the full supercritical range of p > 2(d+k) d , and in particular, we "lose k variables" from the critical exponent 2(d+k) d . Note that the exponent 2(d+k) d + 2k d approximates the critical exponent 2(d+k) d when the dimension d + 1 of the ambient space is large with respect to the degree k of the paraboloid. In our proof, this reflects the fact that the splitting behavior (7.4) of exponential sums dominates for large dimensions. By adapting the difficult argument of Bourgain [3] in a more direct fashion, we can recover the complete supercritical range of exponents, but only for sufficiently small dimensions, of size roughly less than the square of the degree.
The proof of Theorems 1.3 and 1.4 exploits available bounds on one-dimensional Weyl sums of degree k, such as estimates of Weyl type and the Poisson formula on major arcs. The poor quality of known minor arc bounds is the main reason for our relative condition on d and k in Theorem 1.4. It is a curious feature that in dimension d = 1 (say), the approach of Bourgain [3] apparently yields the whole supercritical range for the "sparse" curve (x, x k ). Note that this removes the ε-loss in the restriction estimates of Hu and Li [12,13] for these curves. We remark also that for very large dimensions, the method of proof of Theorem 1.4 also yields restriction exponents, but the range so obtained is much narrower than that of Theorem 1.3.
We make a last remark about the exponent τ in Theorems 1.3 and 1.4, which affects dramatically the quality of full restriction estimates we can obtain as corollaries. In those results, one can in fact take τ to be the largest exponent such that, for all α ∈ T such that there exists q, a ∈ Z with N q N k and α − a For a fixed degree k, the best one can hope for τ is to be 1/k; see e.g. [18,Problem 8,p. 196]. If (1.8) were to hold for all τ < 1/k, then Theorem 1.4 would improve to the full supercritical range in all dimensions. Instead the range in Theorem 1.4 relies on the best known unconditional exponent τ , which is τ = 1 k(k−1) for large values of k by Bourgain-Demeter-Guth's recent resolution of Vinogradov's mean value conjecture [6], or τ = 2 1−k for small values of k by the much simpler Weyl inequality [23] (see Appendix A for more information). Improved bounds on Weyl sums are known for intermediate values of k, but they typically take a different shape than (1.8), and therefore we do not try to incorporate them in our argument. In conclusion, it seems that one current limitation of number-theoretic approaches to restriction estimates for surfaces of high degree is the poor quality of known minor arc bounds for Weyl sums. In fact, even optimal Weyl exponents would not allow us to obtain efficient full restriction estimates. Fortunately, results of ε-removal type ignore minor arcs to some extent, hence the efficient ranges in those cases.

Notation
For functions f : T d → C and g : Z d → C we define the Fourier transforms of f and g by f (k) = T d f (α)e(−α · k)dα and g(α) = n∈Z d g(n)e(α · n). For a function h : For any function f defined on an abelian group, we let f (x) = f (−x). Given a function f : R d → R and two subsets A, B of R d , we write A ≺ f ≺ B when 0 f 1 everywhere, f = 1 on A and f = 0 outside B. We denote the disjoint union of two sets A and B by A B.
When P is a certain property, we let 1 P denote the boolean equal to 1 when P holds and 0 otherwise, and when E is a set we define the indicator function of E by 1 E (x) = 1 x∈E . When p ∈ [1, +∞] is an exponent, we systematically denote by p ∈ [1, +∞] its dual exponent satisfying 1 p + 1 p = 1. We let dm denote the Lebesgue measure on R d , or on T d identified with any fundamental domain of the form [θ, 1 + θ) d , and we let d denote the counting measure on a discrete set such as Z d . When p ∈ [1, +∞] is an exponent and f : However, when α is an element of T or R, then α denotes the distance from α to the nearest integer. For α ∈ T, we let τ α denote the translation operator which maps a function f : For q 2 we occasionally use Z q as a shorthand for the group Z/qZ. When N is an integer we write [N ] = {1, . . . , N }. Throughout the article, we use the letter ε generically to denote a constant which can be taken arbitrarily small, and whose value may change in each occurrence.

Analytic preliminaries
In this section we discuss several standard tools in discrete restriction theory, such as even moment bounds, the epsilon-removal process, and Bourgain's [2,3] discrete version of the Tomas-Stein argument [25,Chapter 7] from Euclidean harmonic analysis.
We will often use a smooth weight function ω : R → [0, 1] of the form Given a dimension d 1, we also define the tensorized version  (3.4) which are the extension operator of our surface S N acting on the sequence a and the Fourier transform of the ω-smoothed counting measure on S 2N , respectively. For any integer s 1, we define R s,P : Z r → C at u ∈ Z r by R s,P (u) = #{ n 1 , . . . , n s ∈ S : P(n 1 ) + · · · + P(n s ) = u }. (3.5) We have the following well-known even moment bound: This observation is occasionally useful to get L 2 → L 2s from bounds on moments of unweighted exponential sums or from arithmetic considerations on the number of representations by a system of polynomials. To see how (3.6) is proven, observe that The second inequality in (3.6) is obtained by orthogonality: From [3], we recall the simple technique by which one eliminates ε-losses in restriction estimates, using a truncated restriction estimate. .
Remark 3.2 Unless otherwise stated, we will apply Lemma 3.1 with C = 0 so that the set {|F a | 0} is the entire torus T r and therefore, the above bounds will apply to Proof We may assume that a 2 = 1 by homogeneity. We have We now discuss the discrete Tomas-Stein argument, which is the starting point of many of our later arguments. We introduce a parameter λ > 0 and define This notation will be reused in later sections. Note that, by Cauchy-Schwarz in (3.3), we always have |F a | C N d/2 (for instance one may take C = 3 d ), and thus we assume that the parameter λ lies in (0, We can view the sequences a(n) and ω d (n) in (3.3) and (3.4) as functions of P(n).
By Cauchy-Schwarz and under the assumption a 2 = 1, it follows that By another application of Parseval, we conclude that We will use this inequality to obtain bounds of the expected order on the level sets E λ . Via the Hardy-Littlewood method, the kernel F may typically be decomposed into a main piece F M and an error term F m corresponding, respectively, to major and minor arcs, and the Tomas-Stein argument reduces matters to obtaining operator bounds for the convolution with F M and demonstrating uniform power saving on F m . This strategy originated in [2,3] and appeared for instance in [11,Section 4] and [27, Section 7] to prove ε-free boundedness of the extension operator applied to the curve (x, x 2 , . . . , x k ); there bounds on moments of F M were used to derive operator norm bounds. The following general lemma abstracts and generalizes this approach, and it shows how to obtain a bound of the form (ii) in Lemma 3.1 from the decomposition we just described.

Lemma 3.3 Suppose that there exists a decomposition F
holds for all q > p with ζ = dτ 2 . Remark 3.4 Scalar variants of this argument existed in many works of the circle method. See for instance Section 7.3 of [23]. See also [6] for a scalar version that closely resembles the one here.
Remark 3. 5 We take a moment to compare this to the Keil-Zhao device in [27], which derives its name from Theorem 4.1 of [15]. The Keil-Zhao device is Tomas's original argument [22] (applied to discrete quadrics instead of the continuous sphere), where Keil writes out the T T * operator into an equivalent expression, before applying Tomas's remarkable insight of decomposing the operator into various pieces and finding appropriate L 1 → L ∞ and L 2 → L 2 bounds.
Proof We assume again that a 2 = 1. By (3.7), Rearranging implies that |E λ | λ − p N dp 2 −K . The result then follows from the layer cake formula and our assumption q > p: The next lemma demonstrates how incorporating subcritical estimates improves the supercritical ranges.

Lemma 3.6 Suppose that
We assume that a 2 = 1. The estimate of (i) at exponent p 1 is also valid for exponents p p 1 (using the trivial bound F a ∞ N d 2 ), therefore it suffices to use the second estimate to bound the tail This has the desired order of magnitude under our condition on p.
This lemma has appeared implicitly in previous work, for example with p 0 = 2 in [3, Eq. (3.111)], or with p 0 = 4 or 6 in [12,13]. In our work, we only use Plancherel's theorem to exploit the subcritical estimate at p 0 = 2.

Restriction estimates for k-th powers
In this section, we obtain truncated restriction estimates for the surface of k-th powers of integers, that is, for (1.6). We fix a degree k 3, and for a sequence a : In this section, we prove the first statement of Theorem 1.1, as follows.
Before embarking on the proof, we derive two consequences of Theorem 4.1 mentioned in the introduction. The first consequence is the second restriction estimate of Theorem 1.1.

Corollary 4.2 Let k 3 and
Proof We use Theorem 4.1 to obtain the first estimate in the assumptions of Lemma 3.6, and the trivial Plancherel estimate at p 0 = 2 to obtain the second one.
Secondly, we obtain the whole supercritical range of restriction estimates under Hypothesis K, by exploiting conjectural estimates for even exponents of lower order.
Proof of Corollary 1.2 Assuming Hypothesis K for k, we have R k,P ∞ ε N ε with P(n) = n k in the notation (3.5), so that by (3.6), F a 2k 2k ε N ε a 2k 2k . By Theorem 4.1 and Lemma 3.1, we deduce that for p > 2k, the following ε-free estimate holds as well: We now set out to prove Theorem 4.1. We fix a sequence a : Z → C supported on [N ] such that a 2 = 1, and a weight function ω of the form (3.1). We let We also introduce a parameter λ ∈ (0, N 1/2 ] and define We recall the Tomas-Stein inequality (3.7) from Sect. 3: We employ the traditional Hardy-Littlewood circle method to understand the magnitude of the exponential sum |F|. We set τ = max 2 1−k , 1 k(k−1) , in accordance with the Weyl-type estimates of Appendix A we intend to use, and we fix a constant δ = kτ + ε. For a parameter 1 Q N δ , we define the major and minor arcs of level Q by We take a few measures to simplify the exposition in the rest of this section. We assume implicitly that N is large enough with respect to k and δ as well as the various ε quantities for the argument to work, without further indication. This is certainly possible since Theorem 4.1 with a 2 = 1 is trivial for N bounded (since |F a | N d/2 ). With these conventions in place, we now obtain two majorants for the exponential sum F on minor and major arcs of level Q, via standard techniques from the circle method recalled in Appendix A.
Otherwise, Proposition A.2 shows that This gives the desired bound if α ∈ M Q , and if α / ∈ M Q , then either q > Q or |β| > Q N k , and in either case We define a majorant function V p,Q : T → C by 2 for 1 Q N δ . While V p,Q is a rather coarse majorant function, it has the advantage that its Fourier transform at nonzero frequencies can be efficiently bounded: in additive combinatorics language, it behaves pseudorandomly. This can be used in turn to obtain efficient L 2 → L 2 bounds for the operator of convolution with V p,Q . This was the approach taken by Bourgain [2] in the case of squares k = 2. We follow this approach and start by bounding the Fourier transform of the majorant V p,Q with the help of the truncated divisor functions d( , Q) = n Q : n| 1.

Proposition 4.4 If p > k, we have
Proof By a linear change of variables, we have By a spherical change of coordinates, we see therefore that Recalling (4.6), this can be used to estimate V p,Q in L 1 when p = k: Performing Fourier inversion in (4.6), we obtain also By orthogonality it follows that The sum inside the parenthesis is bounded by N ε d( , Q) if p = k and by d( , Q) if p > k and ε is small enough with respect to p. Using also Z p ∞ Z p 1 and the estimate (4.11), this concludes the proof.
We begin by removing the minor arcs contribution to the expression (4.4), and we use L p norms to estimate the remaining piece.
We can now derive our first level set estimate, which features an N ε term.

13)
Proof We assume that η N −δ/2k+ε and let Q = η −2k−ε . By Proposition 4.5 with p = k and Fourier inversion, it follows that By Plancherel, (4.9) and (4.10), we obtain We have η N −δ/2k N −1/4+ε , and therefore η 2k Q N ε−k , so that We now obtain a level set estimate designed to remove the N ε that arises in using Proposition 4.9 to bound the moments of F a . We first introduce a technical tool to keep track of the information that the Fourier transform of F has support in [N k ].

Consider a non-negative trigonometric polynomial
By Fourier inversion, we can see that F = F * ψ N . Starting from (4.4), it is then easy to obtain the following analogue of Proposition 4.5.

Proposition 4.7
Suppose that η −2k−ε Q N δ . Then, for p 0, At this stage, we need to import a divisor bound used by Bourgain [2]. (4.14) Proof In the sum of (4.14), the term = 0 contributes at most Q B , and by [2, eq. (4.31)] the other terms contribute at most C ε,B Q ε N k . The conclusion follows from our assumption on Q.
We now proceed to our ε-removal level set estimate.

Proposition 4.9
Let ν ∈ (0, 1] be a parameter. There exists a constant c ν > 0 such that, for p > k, Proof We assume again that Cη −2k−ε Q N δ , and we apply Proposition 4.7 for a fixed p > k. By Proposition 4.7, we have Applying (4.8) and the bound ψ N ∞ ψ N = 1, we deduce that Let q, q ∈ [1, ∞] be a dual pair of exponents to be determined later. Assuming that q ∈ N and Q N k/q , applications of Hölder and Proposition 4.8 furnish Rearranging terms in the above, we find that Choose finally Q = Cη −2k−2ε and q a large enough integer so that q < 1 + ν, and note that we obtain the desired bound for ε small enough. The condition Q N k/q is satisfied for η N −c ν with a certain c ν > 0.
We now prove Theorem 4.1 (with the rescaling a 2 = 1) by integrating the previous level set estimates. There exists c ν > 0 such that, for p > 2k, Proof By the layer cake formula and Proposition 4.6, we obtain where p 2k ensured that the η-integral is log N .
The second estimate is obtained similarly, by invoking Proposition 4.9 in place of Proposition 4.6.
Proof of Theorem 4.1 Remember that δ > kτ was arbitrary, and therefore the parameter ζ in Proposition 4.10 can be given a value arbitrarily close to τ 2 . Fixing ζ as such and taking C := N 1/2−ζ along with 0 < c v < ζ, we can now use (4.16) in Lemma 3.1, to remove the N ε factor in (4.15) for p > 2k which is the desired estimate of Theorem 4.1.

Extending the moment method
The method of the previous section extends to many surfaces, due to its reliance on little number-theoretic information. However, it does not seem to produce truncated restriction estimates in the complete supercritical range for many interesting cases, and therefore we only sketch this class of results.
Fix t 1 and a tuple of integers k ∈ Z t with 1 k 1 < · · · < k t . We consider the monomial curve Define also the maximal degree k = k t and the total degree K = k 1 + · · · + k t . For a sequence a : Z → C supported on [N ] define the following exponential sums associated to S: e(α 1 n k t 1 + · · · + α t n k t ) ( α ∈ T t ), F a (α) = n∈Z a(n)e(α 1 n k t 1 + · · · + α t n k t ) ( α ∈ T t ).
It can be checked that the method of Sect. 4 yields truncated restriction exponents in the range p > 2kt, which is quite far from the full supercritical range p > 2(k 1 +· · ·+k t ) for large values of t. It turns out to be more useful to use a different majorant in that situation. We only describe the main steps of this variant since it was already derived in the case k = (1, . . . , k) in previous work (see [11,Section 4], [27,Section 7]). By the circle method, one can obtain a decomposition of the form where S p and J p are, respectively, the singular series and the singular integral defined by It is known from classical work of Hua [14] and Arkhipov-Chubarikov*-Karatsuba [1, Theorems 1.3, 1.4, 2.4, 2.5], that when k = (1, . . . , k), J p < ∞ for p > K + 1 and S p < ∞ for p > K + 2, while when k = (1, . . . , k) and k 4, J p < ∞ for p > K and S p < ∞ for p > K + 1. Via Lemma 3.3, and writing ρ = τ/2, this gives . . . , k) and k 4, and . . . , k). (This last estimate is the one that was already obtained in [11] and [27]). Note that the above ranges of exponent miss the conjectured ones by two or four variables only.

Arc mollifiers
This section serves to introduce a technical tool, borrowed from Bourgain [3, Section 3] and used in the proof of Theorems 1.3 and 1.4. It consists of a collection of multipliers in the frequency variable α ∈ T, which serves as a partition of unity adapted to the major arcs, that is, the collection of small neighborhoods of rationals with small denominator. We recall the natural bounds on these multipliers and their Fourier transform. Throughout the section we fix an integer k 3, which corresponds to the degree k of the k-paraboloid in Sects. 7 and 8. We fix a smooth bump function κ with [−1, 1] ≺ κ ≺ [−2, 2]. Let N = 2 log 2 N , and for every integer 0 s log 2 N define More importantly, for every dyadic integer 1 Q N , we have We let N 1 = c 1 N , for a small constant c 1 ∈ (0, 1]. It is then easy to check that the intervals are all disjoint. For a dyadic integer Q and an integer 0 s log 2 N , we define the arc mollifier (6.5) so that, by (6.2) and disjointness, We finally define Proposition 6. 1 We have 0 λ, ρ 1 and Proof By (6.3), we can rewrite λ as The proposition follows since we assumed that [− U, the interior of U, and they may be viewed as smooth functions over the torus T, by 1-periodization from the interval U. We will view Q,s alternatively as a smooth function on the torus T or on the real line, but note that for an integer n, Q,s (n) has the same definition under both points of view. For n ∈ Z and an integer Q 1 we define The following useful lemma is due to Bourgain [3]. We include the short proof for completeness.

Lemma 6.2 Let δ x be the Dirac function at x. Then
Proof Let γ (s) = κ − κ(2 · ) for 0 s < log 2 N and γ (s) = κ when s = log 2 N . By (6.1) and (6.5), we can write From Lemma 6.2, we deduce the pointwise bound which is uniform in n ∈ Z. When n = 0 the left-hand side is Q,s dm.

Proposition 6.4
For every ε > 0 and A > 0, we have ρ dm 1, (6.10) Proof Since Q is dyadic, it follows from (6.7) and (6.8) that Since we have chosen N 1 = c 0 N with c 1 small enough, we have ρdm 1 as desired. The bound on ρ is derived from (6.9) in a similar fashion, using also the standard divisor bound d(n, Q) d(n) ε n ε .

Restriction estimates for k-paraboloids of arbitrary dimension
In this section, we obtain truncated restriction estimates for the surface (1.7), for an arbitrary dimension d 1 and degree k 3. For simplicity, we write |x| k = (x k 1 + · · · + x k d ) 1/k for vectors x ∈ R d ; this quantity may be negative when k is odd. Note that the system of polynomials P = (x, |x| k k ) has total degree K = d + k, and therefore the critical restriction exponent is p d,k = 2(d+k) d for the surface (1.7). For a sequence a : The following estimate, a slightly more precise version of the first statement in Theorem 1.3, is the main result of this section. Note that we miss the complete supercritical range by a term of size 2k d , but we obtain a uniform result for all dimensions d and degrees k. We record below the corresponding restriction estimate that can be obtained by bounding the tail of the integral.
Proof We invoke Lemma 3.6. The first assumption is verified with ζ ← dτ 2 − ε for any p 1 > 2(d+2k) d by Theorem 7.1 and the second is verified for p 0 = 2 by Plancherel.
Our argument will make use of Lemma 3.3, whose philosophy borrows from the circle method the paradigm of major arc and minor arc estimates. As such we will split our convolution kernel F, defined in (7.2) below, into major arc pieces and a minor arc piece. On the minor arc piece we will only need some power savings on the trivial bound. We decompose the major arc pieces in a fashion similar to [3], but simpler, and we use the Tomas-Stein method to obtain decent estimates.
We introduce some notation before turning to our proof. We fix integers d 1 and k 3 throughout, on which every implicit or explicit constant throughout is allowed depend. The letter Q will always denote an integer of the form 2 r with r 0. We also fix weight functions ω and ω d of the form (3.1) and (3.2), and we define the exponential sums which may be viewed as Fourier transforms of smoothed surface measures on respectively, for general d and for d = 1. Note that the sum over n ∈ Z d in (7.2) splits and we have Another useful observation is that For each dyadic integer Q and integer s 0 such that 1 Q 2 s , we define a piece of our original exponential sum by F(α, θ ). (7.6) Recall that the weight Q,s is essentially a mollified indicator of the 1 2 s N -neighborhood of the set of rationals with denominator of size Q.
We now define the piece F M of our exponential sum corresponding to the union of all major arcs, and the piece F m corresponding to the minor arcs, by Recalling the decomposition (6.7), this means that We fix a Weyl exponent τ = max 2 1−k , 1 k(k−1) . The minor arc estimates of Appendix A translate into the following statement.
Proof Consider α ∈ U such that ρ(α) = 0. Take 1 a q N k−1 such that (a, q) = 1 and |α − a/q| 1/q N k−1 . It follows from Proposition 6.1 that q > N 1 , for else there exists a dyadic integer Q such that q ∼ Q ⇒ Q N 1 and |α − a/q| 1/Q N k−1 , a contradiction. Therefore we have N q N k−1 and we may apply the bound of Proposition A.1 to each Weyl sum in the product (7.4). By (7.8), we have the following immediate corollary.

Corollary 7.4 We have
We can derive a bound on the piece F Q,s of the exponential sum by appealing to major arc bounds. Proposition 7. 5 We have, uniformly for Q 2 s N , Proof Consider α ∈ U. By (7.4) and (7.6), we have If Q,s (α) = 0, then it follows from (6.6) that there exist 1 a q with (a, q) Proposition 7. 6 We have Proof For any (m, ) ∈ Z d+1 , we have The result now follows from (6.8) and the trivial bound Q,s ∞ Q,s 1 .
From the previous physical and Fourier-side estimates on a major arc piece F Q,s , we immediately deduce L 1 → L ∞ and L 2 → L 2 estimates for the operator of convolution with this piece.

Proposition 7.7 Uniformly for Q 2 s N , we have
Proof First note that for any bounded function W : T d+1 → C, we have Applying these two inequalities to W = F Q,s , and inserting the estimates of Propositions 7.5 and 7.6, we obtain the desired bounds.
Interpolation between the previous convolution estimates gives the following result.
We need to sum this up over the major arcs.

Proposition 7.9
If p > 2(d+k)+2k d , then Proof When p > p 0 , Proposition 7.8 and the triangle inequality yield The sum over the dyadic Q is O(1) for (2 + dp 0 k ) 1 p < d k , which gives the range stated in the proposition.
Proof of Theorem 7. 1 We have a decomposition F = F M + F m which satisfies the estimates of Propositions 7.4 and 7.9. The result now follows from Lemma 3.3, recalling that τ = max 2 1−k , 1 k(k−1) .

Restriction estimates for k-paraboloids of low dimension
In this section, we pursue the study of k-paraboloids of the form (1.7) initiated in Sect. 7, but we aim at obtaining results valid in the complete supercritical range of exponents p > 2(d+k) d instead, under a constraint on the dimension d. The following is the main result of this section, which corresponds to Theorem 1.4. Here F a is defined by (7.1) as before.
Note that lifting this result to a complete restriction estimate via Lemma 3.6 would yield the same result as Corollary 7.2 with a more restrictive condition on d, therefore we do not carry out this process. Our method of proof follows again the numbertheoretic approach of Bourgain [3] for the parabola, this time in a fashion closer to the original. Remarkably, this approach does not break down when using the weaker minor arc estimates available for the Weyl sums (7.3) associated to the k-paraboloid. As in that reference, we first obtain a version of the desired estimate which an extra factor N ε , whose proof is simpler and serves as a blueprint for the more technical ε-free case. We fix at the outset a sequence a : Z d → C supported on [−N , N ] d with a 2 = 1, and we reuse the notation introduced in Sect. 7. In particular we work again with the exponential sums (7.2) and (7.3), and we fix again a Weyl exponent τ = max 2 1−k , 1 k(k−1) .

Bounds on major and minor arc pieces of the exponential sum
For each dyadic integer Q and integer s 0 such that 1 Q 2 s , we define a piece of our original exponential sum by By comparison with the simpler definition (7.6), the second term in the parenthesis ensures that F Q,s satisfies good Fourier bounds at non-zero frequencies. However, there is a trade-off in the sense that we only get acceptable physical-side bounds on F Q,s for suffficiently small dimensions, as the next proposition shows.
Proof From the definitions (7.6) and (8.1), we have By Propositions 7.3 and 7.5, and inserting the bounds (6.8) and (6.10), we obtain Since Q 2 s N and (k − 2)/(k −1 − τ ) > d, the second term in the last line may be absorbed into the first for ε small enough.
In the rest of this section, we assume that the hypothesis d < k 2 −2k 1−kτ of Theorem 8.1 is satisfied to avoid repetition. We also introduce a technical device analogous to that of Sect. 4 to ensure that all Fourier transforms under consideration stay inside an N × · · · × N × N k box. We fix a trigonometric polynomial ψ N on T d+1 such that which in particular implies that T d+1 ψ N = 1. When H : T d+1 → C is a bounded measurable function, we writeḢ = H * ψ N for brevity; note that Ḣ p H p for any p 1 by Young's inequality, and that F =Ḟ by (7.5) and Fourier inversion. With this notation in place, we derive a Fourier estimate improving on that of Proposition 7.6, by exploiting the pseudorandomness of the weight Q,s − Q,s ρ ρ.
In particular, we have Proof Let Q,s = Q,s − Q,s ρ ρ and note that Q,s (0) = 0. By a computation similar to that in Proposition 7.6, we find that for any (m, ) ∈ Z d+1 , It then suffices to insert the estimates (6.9) as well as (6.8), (6.10) and (6.11).
We again define a piece F M of our exponential sum corresponding to the union of all major arcs, and a piece F m corresponding to the minor arcs, this time by Proposition 8. 4 We have Proof Recalling the definitions (8.1) and (6.7), we have From (6.8) and (6.10), we deduce that It remains to insert the bound of Proposition 7.3 to conclude the proof.
The previous estimates on F Q,s yield bounds for the operator of convolution with this kernel.

Proposition 8.5 Uniformly for Q 2 s N , we have
Proof By the same argument as in Proposition 7.7, inserting the estimates of Propositions 8.2 and 8.3 instead, the proposition follows.
Interpolation at the critical exponent almost completely removes the operator constant, as the next proposition shows.
In particular, for p = p 0 we have Proof Fix parameters p ∈ (1, 2] and θ ∈ (0, 1] such that By interpolation between the estimates of Proposition 8.5, we obtain Since θ = 2 p , we see that 1 − k+d d θ = 1 − p 0 p , which yields the desired estimate.

ε-Full restriction estimates
In this subsection we derive the upper bound in Theorem 8.1 up to a factor N ε . We fix a weight function a : Z d → C supported in [−N , N ] d , and we may assume without loss of generality that a 2 = 1 in proving that variant of Theorem 8.1. We introduce the usual level set E λ and weighted indicator f defined by Proof Starting from (8.9), and using the triangle and Hölder's inequalities, we obtain By (7.10) and (8.7), it follows that Assuming that λ N d/2−dτ/2+ε , we infer that The previous level set estimate may be integrated into a truncated ε-full restriction estimate.
Proof It suffices to invoke Proposition 8.7 in

ε-Free restriction estimates
The goal of this section is to derive Theorem 8.1 in full. While we use propositions from the previous subsection, we do not need the final ε-full estimate of Proposition 8.8. We start by stating a distributional version of Lemma 4.8 (which follows immediately from Markov's inequality).
Proof Consider the real number θ ∈ (0, 1) such that (8.8) holds. By convexity of L p norms, we have Applying (8.4), and (8.10), we obtain Since | f | takes values in {0, 1}, we may rewrite this as The proof is finished upon observing that θ = 2 p by (8.8), and recalling that p 0 = 2(k+d) d .
Following the argument of Bourgain [3], we distinguish two cases according to the size of Q. We introduce a parameter Q 1 1, and we write F M = F 1 + F 2 with Choose D = η −ν for parameter ν > 0, so that Choosing B C/ν with C > 0 large enough, we deduce that |E λ | N −(k+d) η − p −ν .
Since q := p + ν can be chosen arbitrarily close to 2(k+d) d , this concludes the proof, upon recalling that η = λN −d/2 .
On the major arcs, we use a majorant obtained through the Poisson formula and standard bounds on oscillatory integrals and Gaussian sums. We write J (β, θ − b q − m; N ) = R η(x)e (N φ b,m (x))dx, where On the support of η, we have |x| 2 and therefore under our size condition on β. We fix a large enough constant C > 0. The second term may be absorbed into the first since |β| 1 q N k−1 and 1 q N , and this concludes the proof.