Papers

We study the Assouad and lower dimensions of graphs of Weierstrass-type functions of the form \[ f(x)=\sum_{n=1}^{\infty}a_n\phi(b_nx+\theta_n), \qquad x\in\mathbb R, \] where $\phi$ is a nonconstant $C^2$ function of period one, $a_n,b_n>0$, $(\theta_n)$ is an arbitrary sequence of real numbers, $\sum_{n=1}^{\infty}a_n<\infty$, and $b_{n+1}/b_n\to\infty$. We give quantitative conditions under which the graph over every nondegenerate compact interval has Assouad dimension two or lower dimension one. In particular, when $a_n=b_n^{-\alpha}$, we obtain explicit frequency-gap conditions under which the Assouad spectrum of the graph is equal to $2$ for $\alpha\leq\vartheta<1$. When the logarithmic frequency ratios converge, we also determine the spectrum on a nonempty interval below $\alpha$. Together with Barański's formulas for the Hausdorff and box dimensions, these results yield graphs whose lower, Hausdorff, upper box, and Assouad dimensions are four distinct numbers.

We obtain optimal Hardy inequalities for fractional powers of the $p$-Laplacian $\Delta_p^\sigma$ on $\mathbb{Z}$, $p\in (1,\infty)$, $\sigma \in (0, 1/p)$. Asymptotically, the optimal weights behave like $|x|^{-p\sigma}$ as $|x|\to \infty$. We show a similar result for the Riesz fractional $p$-Laplacian. In an appendix, we show that for general $p$-Laplace operators on locally summable graphs, $p$-null-criticality implies $p$-optimality at infinity.

We study optimal polynomial approximants in spaces of Dirichlet-type over the bidisc. Given a nonzero function f, the linear optimal polynomial approximant (opa) to its reciprocal is the affine polynomial p for which pf is closest to the constant function 1 in the space norm. We prove that, for every positive value of the Dirichlet-type parameter, there exists a symmetric polynomial f with no zeros on the closed bidisc, while the corresponding linear optimal approximant has a zero inside the bidisc. This provides polynomial counterexamples to the version of the Weak Shanks Conjecture on these spaces. The construction also extends to higher-dimensional polydiscs and to anisotropic Dirichlet-type spaces.

We study the approximation of finite-energy Sobolev functions by bounded Sobolev functions for integral functionals associated with possibly signed Carathéodory integrands. The approximating sequence is required to converge strongly in the Sobolev space, while the corresponding energy densities converge in \(L^1\). Ordinary truncations may fail because they replace the gradient by zero on the tails, where the integrand need not be integrable. Our main contribution is a general method based on spatially dependent barriers whose gradients can be adapted to the structure of the Lagrangian.

We study the computational cost and stability of positive cubature and hyperinterpolation on a truncated conic surface with a Jacobi-type weight. A classical quadratic disk-to-cone map is used in a basis-free form as a measure-preserving $\mathbb Z_2$ quotient. It identifies the full degree-$m$ conic trace space with the even disk polynomials of degree at most $2m$ and converts positive degree-$2n$ cone cubature into centrally symmetric positive degree-$(4n+1)$ disk cubature, and conversely. This yields an exact transfer of Möller's lower bound and of the node excess above it, so that near-minimal disk formulas produce compressed non-product cone rules. The same quotient transfers reproducing kernels and hyperinterpolation operators. Every positive degree-$2n$ cone rule gives an exact $L^2$ sampling isometry on $\Pi_n(V)$; in particular, the weighted sampling matrix has condition number one, independently of the number and geometry of the nodes. For the unweighted radial case $\gamma=0$, if $\Lambda_n^V$ denotes the $C(V)\to C(V)$ Lebesgue constant of degree-$n$ hyperinterpolation, then every positive degree-$2n$ cone cubature rule satisfies the rule-independent sharp law $$c n \le \Lambda_n^V \le C n.$$ More generally, the upper bound $\Lambda_{n,\gamma}^V \le C_\gamma n^{\gamma+1}$ holds when $\gamma$ is a nonnegative half-integer. Thus node compression preserves exact $L^2$ conditioning while the associated $L^\infty$ stability has the optimal universal order. Low-degree near-minimal rules and higher-degree optimized disk schemes illustrate the reduction in sampling cost.

We extend Rainwater's theorem that, for Baire measures on a compact Hausdorff space,a measure singular with respect to any measure from a $w^*$-compact convex set $M$ of probability measures sits on a Baire set which is $\mu$-null for every $\mu\in M$. Our main result shows that this statement holds under the weaker assumption of convex analyticity of $M$. In the standard way, this extends the Glicksberg–König–Seever decomposition theorem. We also give conditions for our results to hold for sets of signed measures.

We develop a general criterion for establishing absolute continuity of convolutions of fractal measures on the line, and more generally of prescribed line projections of fractal measures in the plane. The crucial new ingredient is exact scaling covariance of the \(L^1\)-norm of Littlewood-Paley pieces of the projected measure, under affine renormalization. We apply this criterion in three key settings. First, we show that the convolution of two measures on the line is absolutely continuous whenever their dimensions sum to more than one, provided one is a self-conformal measure whose defining IFS is not \(C^2\)-conjugate to linear, and the other is either self-conformal or Ahlfors–David regular. Second, we show that every line projection of a planar complex-analytic self-conformal measure of dimension greater than one is absolutely continuous, under natural nonlinearity and nondegeneracy assumptions. Finally, we show that every line projection of a planar self-affine measure is absolutely continuous, under natural irreducibility and proximality assumptions, whenever the correlation dimension of the measure and the Frostman dimension of its Furstenberg measure sum to more than two.

Let $d \geq 5$, $0 < \gamma < d - 2$, and $\Omega_N$ be the binomial random subset of $Q_N = [-N,N]^d \cap \mathbb Z^d$ with retention probability $p_N = N^{-\gamma}$. We prove that, with failure probability of optimal exponential order, every subset $B \subseteq \Omega_N$ of fixed positive relative density realizes, at each scale $p_N^{-2/(d - 2)} \lesssim \lambda \lesssim N^2$, a squared distance of the form $q^2 \lambda$, where $q$ belongs to a fixed finite set depending only on the dimension and the density. The lower scale $p_N^{-2/(d - 2)}$ is sharp. As a consequence,the squared-distance set $D^2(B)$ of $B$ has maximal order $N^2$ and contains affine copies of every fixed finite subset of $\mathbb Z$. The main new input is a finite multidilate supersaturation theorem for dense subsets of $Q_N$, which, together with boundedness estimates for the associated spherical distance graphs down to the sharp scale, allows us to apply Schacht's transference theorem.

We prove a formula for the total variation of SBV functions in sub-Riemannian Heisenberg groups arising as the limit of certain BMO-type functionals.

We prove a Jackson-type inequality for approximation by a prescribed number of terms of a band-limited biorthogonal wavelet system in homogeneous weighted Besov spaces with Muckenhoupt weights, on functions recovered from their wavelet expansion along the dyadic tree of the unit cube. The error decays at a rate set by the smoothness-to-dimension ratio. The explicit constant has a closed-form majorant, and the weight enters only through a displayed power of the Muckenhoupt characteristic. An extremal family shows that the exponent of this rate cannot be increased on the class, and an unweighted example shows that the localization is necessary.

Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. It follows from a theorem of Bohr that if $f$ in $\mathscr{H}^2$ has a bounded analytic continuation to the right half-plane, then its norm can be computed from the mean values \[\|f\|_{\mathscr{H}^2}^2 = \lim_{\sigma\to 0^+} \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T \lvert f(\sigma+it) \rvert^2\,dt.\] We investigate to what extent the assumption that the analytic continuation be bounded can be relaxed, and obtain optimal results in two different directions. The first direction concerns the order of the analytic continuation, and yields as a corollary a version of the classical Landau–Schnee convergence theorem for $\mathscr{H}^2$. The second direction concerns the abscissa of convergence, and resolves a problem posed by McCarthy.

Pólya and Szegő conjectured in 1945 that the disk achieves the largest electrostatic capacity among all planar sets with given logarithmic capacity. This question generalizes naturally to balls and Newtonian capacities in higher dimensions. The conjecture remains open. This paper establishes a conditional result: the conjecture would follow if among all lower dimensional sets in a horizontal strip with Neumann boundary conditions, the transverse component of the Dirichlet integral of the equilibrium potential could be shown minimal for the lower dimensional ball. As supporting evidence toward this latter property, the paper shows extremality of the ball for $L^p$-norms of the potentials, via Baernstein star-function symmetrization techniques.

We construct the first convex planar counterexample to the Schiffer and Pompeiu conjectures: a bounded strictly convex non-disc domain with real-analytic boundary carrying a nonconstant solution of $\Delta u+u=0$ with $u=1$ and $\partial_\nu u=0$ on the boundary. Together with the higher-dimensional constructions, this gives convex non-ball counterexamples in every dimension from two to eighteen, and in dimensions twenty and twenty-one, with real-analytic boundaries diffeomorphic to spheres and indicator Fourier transforms vanishing on the unit sphere. In dimensions $4$, $6$, $8$, $10$ and $14$, planar reductions through compact Lie algebras of rank two use Harish-Chandra's radial part formula and Kostant's convexity theorem. In dimensions $3$, $5$ and $7$, an axisymmetric formulation on the unit ball gives a quartic equation in weighted coefficient spaces. The exact two-sided inverse of the residual-corrected linearisation combines a Dirichlet Helmholtz inverse, a holomorphic boundary real-part problem and the material-derivative identity; it gives the planar Schiffer domain and, by one dimension-parametrised argument, the domains in dimensions $5$, $9$, $11$ to $13$, $15$ to $18$, $20$ and $21$. We also construct the first convex planar counterexample to Berenstein's conjecture: a strictly convex non-disc domain with real-analytic boundary carrying a sign-changing solution of $\Delta u+u=0$ with $u=0$ and $\partial_\nu u=1$ on the boundary. Planar existence for this problem is due to Colbrook, Sadeghi and Stepaniants. All existence proofs in this paper are computer-assisted.

We show that an almost-everywhere upper bound $b\geq 1$ for the Hausdorff dimension of visible parts of all compact planar sets implies the upper bound $d-2+b$ for visible parts of all compact sets $K$ in $\mathbb{R}^d$, $d\geq3$. Inserting the sharp planar bound $b=3/2$ of Orponen and Rutar \cite{OrponenRutar2026} gives $\dim_H \mathrm{Vis}_e(K)\leq d-1/2$ for almost every $e\in S^{d-1}$. This would answer their Question 1.9 sharply in all dimensions.

STFT phase retrieval aims to reconstruct a signal from spectrogram magnitudes. While two compactly supported windows theoretically guarantee uniqueness, a complete algorithmic framework remains lacking. This paper systematically constructs numerical schemes for multi-window STFT phase retrieval. We theoretically provide a nonuniqueness counterexample for the single-window case. We then propose a two-window reconstruction algorithm based on Shannon interpolation and analytic square-root extraction. To overcome the instability of the global sign choice, we introduce a third window and a robust $K$-sign method that determines the sign locally. We also adapt SCR algorithm to the STFT setting. For the $K$-sign method, we establish deterministic stability on a finite contiguous block of frequency samples, with the contribution of the complementary tail explicitly included in the error bound. Numerical experiments validate the effectiveness of all schemes.

We prove a sharp Szemerédi-Trotter estimate \[\mathcal{I}(A,\mathcal{L})\lesssim \frac{|A||\mathcal{L}|}{p}+|A|^{2/3}|\mathcal{L}|^{2/3}+|A|+|\mathcal{L}|\] for prime finite field $\mathbb{F}=\mathbb{F}_p$ by a new polynomial decomposition theorem. As applications, we first prove the sharp Furstenberg set estimate in $\mathbb{F}^2$. Secondly, we improve sum-product estimate \[\max\{|A+A|,|A\cdot A|\}\gtrsim\min\{(p|A|)^{1/2},|A|^{5/4}\},\quad A\subset\mathbb{F}.\] Finally, we improve the Fourier restriction estimate $R^*(2\to\alpha)$ holds for $\alpha>\frac{10}{3}$ in $\mathbb{F}^3$ when $p\equiv 3\mod 4$.

Let $d \ge 2$, let $\alpha_1,\dots, \alpha_n > 0$, and put $A = \sum_k \alpha_k$. We determine, up to constants depending only on $d$, the smallest possible $L^1$ norm in the unit ball $\mathbb{B}^d\subset \mathbb{R}^d$ of the Coulomb field generated by charges of strengths $\alpha_k$ placed on the unit sphere $\mathbb{S}^{d-1}$. More precisely, we prove \[ \inf_{x_1,\dots,x_n \in \mathbb{S}^{d-1}} \int_{\mathbb{B}^d} \left| \sum_{k=1}^n \alpha_k \frac{x_k-x}{|x_k - x|^d} \right| dx \asymp_d A^{-1/(d-1)} \sum_{k=1}^n \alpha_k^{d/(d-1)}. \] The lower bound follows from an $L^1$ Lipschitz trace estimate and an explicit weighted tent function on the sphere. For the upper bound we use a partition of the sphere into cells of masses $\alpha_k/A$ and diameters $O_d \left((\alpha_k/A)^{1/(d-1)} \right)$. For the unit weights the sharp order is $n^{(d-2)/(d-1)}$, and in dimension three it is $\sqrt{n}$.

In this work, we provide a complete characterisation of Pitt's inequality for metaplectic operators. The symplectic geometry underlying a metaplectic operator distinguishes effective directions, along which its action is Fourier-type, from singular directions, along which concentration is preserved. For this reason, we adopt two complementary perspectives, thereby obtaining both an isotropic Pitt's inequality, emulating the classical theorem of Pitt, and an anisotropic inequality that adapts to the geometric features of the metaplectic group. As a by-product, we obtain Pitt's inequality for Fourier transforms along subspaces of $\rd$ and new quantitative time-dependent boundedness results for metaplectic operators on homogeneous Sobolev spaces, with applications to Schrödinger evolutions generated by quadratic Hamiltonians. Additionally, we derive logarithmic and entropic uncertainty principles for the metaplectic group. We show that the classical entropic uncertainty principle fails when both the effective and singular directions are present. In this case, a generally unbounded corrective entropy term accounts for the concentration-preserving directions and restores a meaningful lower bound. For quadratic Schrödinger evolutions, this correction follows the time-dependent geometry of the nondispersive directions and adjusts the uncertainty estimate accordingly.

For functions that are convex of higher order, classical extremalities identify Gaussian quadrature as a lower bound and Gauss–Lobatto quadrature as an upper bound for the integral. We sharpen this bracket in every odd order: if $f$ is $(2n-1)$-convex on $[-1,1]$, then the exact integral $I[f]:=\int_{-1}^{1}f(x)\,dx$ lies not merely between the $n$-point Gauss–Legendre rule $G_n[f]$ and the $(n+1)$-point Gauss–Lobatto rule $L_{n+1}[f]$, but already between $G_n[f]$ and their midpoint. This yields the certified estimator $Q_n[f]:=\frac{3}{4}G_n[f]+\frac{1}{4}L_{n+1}[f]$, with $|I[f]-Q_n[f]|\leqslant\frac{1}{4}|L_{n+1}[f]-G_n[f]|$, and the constant $\frac{1}{4}$ is sharp. We also record a complementary phenomenon in even order: for $2n$-convex functions the Gauss–Radau endpoint rules still bracket the integral, but the midpoint of the two Radau rules cannot yield a one-sided refinement. The obstruction is explained by an odd, sign-changing Radau quadrature kernel. Finally, we distinguish numerical certification from ordinary high-order approximation: we include concrete examples from spline, moment and statistical models and report experiments for truncated powers and near-pole Stieltjes kernels, where the shape certificate remains meaningful when classical smoothness-based error constants are unavailable or severely pessimistic.

We study weak tiling by finite unions of pairwise non-overlapping unit cubes in $\mathbb R^d$ that are not necessarily axis-parallel. We obtain a weak-tiling analogue of Keller's classical theorem, which gives a structural restriction on the support of any weak tiling measure associated with the unit cube. This allows us to derive geometric restrictions on configurations of cubes whose union admits a weak tiling. As an application, we prove Fuglede's conjecture for unions of three non-overlapping unit squares in $\mathbb R^2$ as well as unions of two non-overlapping axis-parallel unit cubes in any dimension.

We study wave packet density estimates in the local smoothing problem for the wave equation. Building on the framework of Gan, He, Li, and Wu, we refine the broad-incidence argument to obtain an improved local smoothing range in even dimensions. We then investigate the limitations of wave packet density methods, showing that genuine geometric obstructions already arise in the Euclidean setting.

For integers $n>k\geq0$ and $1\leq p,q\leq\infty$, let $\lambda_3(n,k,p,q)$ denote the optimal constant in the one-dimensional Sobolev inequality \[ \|u^{(k)}\|_{L^q(0,1)} \leq \lambda_3(n,k,p,q) \|u^{(n)}\|_{L^p(0,1)}, \qquad u\in\mathring W_p^n(0,1). \] Nazarov and Shcheglova \cite{NazarovShcheglova} conjectured that \[ \lambda_3(n,1,p,1) = 2\lambda_3(n,0,p,\infty), \qquad n\geq2,\quad 1\leq p\leq\infty, \] and that the corresponding extremal functions coincide and are symmetric about the midpoint of the interval. We prove this conjecture for $(n,p)=(3,2)$ and, in particular, obtain \[ \lambda_3(3,1,2,1) = \frac{1}{32\sqrt5}. \] The proof is based on a reduction to an operator norm problem with an additional moment constraint. After rescaling to $(-1,1)$, this constraint leads to orthogonality with respect to quadratic polynomials. We extend the inverse of the second derivative through the corresponding orthogonal projection and analyze the adjoint operator by an odd–even decomposition. The odd component is controlled by a positive Gram kernel, while the even component is treated by a weak-$L^2$ estimate. The equality cases yield the characterization of all extremal functions.

For $\alpha\in(0,n)$, $p\in(1,\frac{n}{\alpha})$, $q:=\frac{np}{n-\alpha p}$, and $W\in\mathscr A_{p,q}$, we obtain the following matrix-weighted boundedness of the fractional integral $I_\alpha$: for any $\vec{f}\in L^p(W^p)$, \begin{align*} \left\|I_\alpha\vec f\right\|_{L^q(W^q)} \lesssim[W]_{\mathscr A_{p,q}}^{ (1-\frac{\alpha}{n})\max\{1,\frac{p'}{q}\}} \left\|\vec f\right\|_{L^p(W^p)}, \end{align*} where the implicit positive constant is independent of $W$ and $\vec{f}$. For integer $n\geq2$, $p\in[1,n)$, $q:=\frac{np}{n-p}$, and $W\in\mathscr A_{p,q}$, we establish the following matrix-weighted Sobolev inequality: for any smooth $\mathbb{C}^d$-valued function $\vec{f}$ with compact support, \begin{align*} \left\|\vec{f}\right\|_{L^{q}(W^q)} \lesssim[W]_{\mathscr A_{p,q}}^{\frac{n-1}{n}} \left\|WD\vec{f}\right\|_{L^p(\mathbb R^n,\mathbb C^{d\times n})}, \end{align*} where the implicit positive constant is independent of $W$ and $\vec{f}$ and where $D\vec{f}$ is the Jacobian matrix of $\vec{f}$. Surprisingly, in both bounds above the exponents of $[W]_{\mathscr A_{p,q}}$ coincide with the corresponding known optimal scalar exponents, and hence are optimal, which is different from the Calderón–Zygmund operator case. A key idea to obtain the above matrix-weighted boundedness of the fractional integrals is to establish the off-diagonal chain-packing principle of dyadic cubes, which is inspired by a recent work of A. K. Lerner.

Let $T$ be the classical Takagi function, and $L(y):=\{ x \in [0,1] : T(x) = y \}$ be its level set at height $y$. For each positive integer $m$, let $S_m$ be the set of $y$ for which $L(y)$ has exactly $m$ points. We prove that $S_{2n}$ has positive Lebesgue measure for every positive integer $n$, confirming a conjecture of Allaart.

Let $d\geq 2$ and $T^*_{\Omega,b}$ be the commutator of the rough maximal singular integral on $\mathbb{R}^d$ defined by $$T^*_{\Omega,b}f(x)=\sup_{\varepsilon>0}\left|\int_{|x-y|>\varepsilon}\bigl(b(x)-b(y)\bigr)\frac{\Omega(x-y)}{|x-y|^d}f(y)\,dy\right|.$$ Assume that $\Omega\in L(\log L)^2(\mathbb{S}^{d-1})$ has mean zero and that $b\in\operatorname{BMO}(\mathbb{R}^d)$. We prove that, for every $\lambda>0$, \[\bigl|\{x\in\mathbb{R}^d:T^*_{\Omega,b}f(x)>\lambda\}\bigr|\lesssim_{\Omega,b}\int_{\mathbb{R}^d}\frac{|f(x)|}{\lambda}\log\!\left(e+\frac{|f(x)|}{\lambda}\right)\,dx.\] The argument starts from a Calderón-Zygmund decomposition of $f$ and a dyadic linearization of the maximal truncation. A decomposition of $\Omega$ by size, followed by regularization of the spatial cutoffs and a microlocal decomposition, reduces the problem to a family of estimates with quantitative decay. The required decay follows by interpolating localized $L^1$ and $L^3$ bounds and using an analytic-family argument for the commutators.

We give an affirmative answer to the planar local smoothing problem in Question~1.7 of D.~Ntalampekos and positive and negative answers to the basic approximation and level-set parts of his higher-dimensional Question~1.8. In every dimension $n\ge2$, each continuous Lebesgue monotone function in $W^{1,p}$ on a bounded open set admits uniform and strong $W^{1,p}$ approximation by monotone $C^{1,\alpha}_{\loc}$ functions, with unchanged Sobolev boundary values and no increase of the $p$-Dirichlet energy, for $1<p<\infty$. The energy can be made strictly smaller unless the original function is $p$-harmonic. In the plane we obtain smooth local replacement at every isolated $p$-harmonic critical point, with arbitrarily small $C^1$ and Sobolev error. A point of gradient index $-m$ can be resolved into exactly $m$ nondegenerate saddles. Together with Ntalampekos's planar theorem, this gives smooth monotone density with fixed boundary values; at a nonsmooth $p$-harmonic minimizer the energy increase is unavoidable but can tend to zero. In dimensions $n\ge3$, an explicit Lipschitz monotone function has a nonmanifold point on every level in an interval, although it admits smooth monotone approximation. A product extension of a planar homogeneous $7$-harmonic function also rules out a general discrete exceptional set under the exact boundary and energy constraints. The answer to Question~1.8 thus distinguishes the five basic approximation properties from the stronger topological and exceptional-set conclusions. Applications include constrained integral functionals, strict $BV$ convergence of superlevel sets, nonlinear flux convergence, and stability of persistence diagrams under tameness assumptions.

We develop a hypergeometric framework for elliptic integral moments organized around Mishev's completed Saalschützian $L$-function and its $W(D_5)$ symmetry. The Bailey–Mishev very-well-poised ${}_7F_6(1)$ representation places complementary elliptic integral moments and harmonic hypergeometric sums in a common parameter space. We first give collision-based hypergeometric derivations of the three complementary-modulus moment families involving $K'^2$, $E'K'$, and $E'^2$. The $E'^2$ identity is also recovered independently from a Barnes-contiguous relation mirroring the differential system of the elliptic integrals. We then derive a hypergeometric evaluation of the harmonic-series representation of the four-Bessel moment $s_{4,0}$. At symmetric parameter points, Coxeter symmetry forces selected Taylor coefficients to vanish along symmetry-adapted paths, producing families of harmonic and special-value identities involving values of the Riemann zeta function and the same Bessel period. The resulting picture shows that these elliptic-moment, harmonic-sum, and special-value identities arise from a common $W(D_5)$ symmetry of generalized hypergeometric functions.

We study logarithmic complete monotonicity of a power family built from the digamma function. An explicit Laplace representation of its negative logarithmic derivative reduces the question to positivity of a continuous kernel. We prove a strict kernel bound on an explicit parameter range and obtain uniform lower bounds for all alternating logarithmic derivatives. In particular, this gives an independent proof of the property conjectured by Guo, Zhang, and Qi on the entire positive half-line. We also characterize the exact admissible parameter threshold by a variational minimum and show that the explicit sufficient bound is not optimal. The argument combines monotonicity of an auxiliary digamma kernel with a regularized logarithmic convolution. An appendix identifies a false auxiliary inequality in an earlier proof, while the present argument independently establishes its stated conclusion.

We study the growth of the spectral function of the Laplace-Beltrami operator on a Riemannian manifold $M$ of dimension $n$. For $n\geq 3$ we show that for any $x\in M$ there is a full measure subset $Z_x\subset M$ such that for all $y\in Z_x$ \[E_{\Lambda}(x,y) = O_{M,x,y,\varepsilon}(\Lambda^{n/2}(\log\Lambda)^{3/2}(\log\log\Lambda)^{1/2+\varepsilon}). \] For $n=2$ we prove a stronger statement. For any $x\in M$ there is a full measure subset $Z_x\subset M$ such that for all $y\in Z_x$ \[E_{\Lambda}(x,y) = O_{M,x,y,\varepsilon}(\Lambda^{5/6}(\log\Lambda)^{1/6}(\log\log\Lambda)^{1/6+\varepsilon}) .\]

We demonstrate a modification of the high-low method of Guth-Maldague-Wang that yields a decoupling theorem which gives the sharp estimate for the discrete restriction constant for the parabola and also a diameter-free bound for the three-fold additive energy of rational points on the parabola and circle. These results were recently obtained by Deng-Fan-Guo-Guo-Luo and Cushman-Demeter-Wu, respectively. The novel features in this argument are an orthogonal expansion of $F^3$ and a new square function that also keeps the wavepackets discarded after pruning, resulting in new and more efficient high and low lemmas.