Papers

An addition formula is a closed-form formula for reproducing kernels of orthogonal polynomials of several variables; the latter are kernels of orthogonal projection operators. We derive addition formulas for two families of orthogonal polynomials, which consist of wrapped products of classical Gegenbauer polynomials on the unit ball with either classical Jacobi polynomials on the simplex or product Laguerre polynomials, each of which is given explicitly as a multilayer of integrals of a classical orthogonal polynomial of one variable. These formulas provide a powerful tool for analysis and allow us, in particular, to reduce a substantial portion of analysis on the Fourier orthogonal expansions on the domains to that of one variable.

A separable metric space has strict measure zero if its packing measure induced by any gauge is zero. We provide a few characterizations of strict measure zero, look at cartesian products and prove that in every Polish locally compact group every set of strict measure zero is null-additive and that in the Cantor set, Euclidean spaces and tori the two notions are equivalent.

We characterize compactly supported functions $f$ in Dunkl–BMO spaces as those that can be written as \begin{equation*} f=g+\widetilde{S}_{\mu,\mathcal{P}}, \end{equation*} the sum of some bounded function $g$ and the generalized balayage $\widetilde{S}_{\mu,\mathcal{P}}$ of a Carleson measure on $\mathbb{R}^{N}\times(0,\infty)$ associated with the Dunkl–Poisson semigroup. As a direct application of this representation of Dunkl–BMO functions, we establish a new characterization of the Hardy space in the Dunkl setting via a principal value integral related to $\widetilde{S}_{\mu,\mathcal{P}}$. Moreover, analogous conclusions hold for the Dunkl–heat semigroup as well.

We give an alternative proof of the $5/4$ result of Guth-Iosevich-Ou-Wang on the Falconer distance conjecture: if a compact set $E$ in the plane has Hausdorff dimension $>5/4$, then there exists $x\in E$ such that the pinned distance set $$\Delta_x(E):=\{|x-y|: y\in E\} $$ has positive Lebesgue measure. Moreover, we are able to beat the $5/4$ barrier if the packing dimension of $E$ is $<2$. More generally, we obtain sufficient conditions involving both the Hausdorff and packing dimensions of $E$. These conditions improve the $5/4$ threshold when the packing dimension of $E$ is $<2$, and recover the second author's result for $\dim_H E=\dim_P E>1$. The proof is based on $L^2$ ball inflation and does not use decoupling inequalities.

Let $d\ge3$ and $2\le r<\infty$. We consider the strong maximal operator associated with Euclidean surface averages over the $\ell^r$ unit sphere and independent coordinate dilations. We prove that this operator is bounded on $L^p(\mathbb R^d)$ if and only if \[ p>\max\left\{\frac{d+1}{d-1},\frac{r}{d-1}\right\}. \] We also obtain $L^p\to L^q$ bounds for the corresponding local maximal operator. For each fixed $p$ in certain ranges, the resulting range of $q$ is optimal up to endpoints. The proof uses a dyadic surface decomposition and frequency-localized $L^2$ estimates. We establish these estimates by combining exact multiplier factorization via iterated stationary phase with scalar Fourier inversion and $TT^*$ arguments. In dimension three, these estimates are combined with quantitative multiparameter local smoothing.

Let $Q_0:=(0,1)^n$. We define $\mathcal{B}_q(Q_0)$ by replacing the local $L^1$ mean oscillation in the Bourgain–Brezis–Mironescu space $\mathcal{B}(Q_0)$ with its $L^q$ counterpart, which coincides with $\mathrm{BMO}(Q_0)$ when $n=1$. For $n\geq2$, our results reveal different endpoint behaviors for general functions, characteristic functions and the exact perimeter formula. (i) For general functions, $\mathcal{B}_q(Q_0)=\mathcal{B}(Q_0)$ precisely when $q\in(0,\frac{n}{n-1})$. We also establish the corresponding sharp self-improvement and duality results for Hardy-type preduals. (ii) For characteristic functions, the endpoint is attained: the equivalence $\|\mathbf{1}_E\|_{\mathcal{B}_q(Q_0)}\sim\|\mathbf{1}_E\|_{\mathcal{B}(Q_0)}\sim\min\{1,P(E,Q_0)\}$ holds uniformly over measurable $E\subset Q_0$ precisely when $q\in(0,\frac{n}{n-1}]$, where $P(E,Q_0)$ denotes relative perimeter. (iii) The isotropic functional $I_{\varepsilon,q}^{*}(E)$, defined using arbitrarily oriented cubes, converges to $\frac{1}{2}\min\{1,P(E,Q_0)\}$ as $\varepsilon\to0^+$ throughout $q\in(0,\frac{n}{n-1}]$. In dimension two, the formula extends to $q\in(0,q_*]$ for some universal $q_*>\frac{19}{9}$, whereas it fails in every dimension for every $q>q_0$, where $q_0>2$ is defined variationally and numerical computation suggests that $q_0\approx2.15391$. The proofs of the three sharp self-improvements use sparse median domination, atomic block decompositions, and relative isoperimetry, respectively. For the perimeter formula, a uniform De Giorgi blow-up reduces the problem to a geometric cube-section inequality; geometric averaging and an explicit packing yield the planar extension and the obstruction, respectively.

Benjamini asked whether every configuration of points on the real line that is in equilibrium under the inverse-square repulsive force must be an arithmetic progression. Georgakopoulos and Kolountzakis proved this when some gap between consecutive points has maximal or minimal length, and described the general (aperiodic) case as open. We show that the answer is yes. More generally, let $1<s\le2$, and let $X\subset\mathbb{R}$ be a locally finite set with at least two points such that, for every $x\in X$, the total force $\sum_{y\in X\setminus\{x\}}|y-x|^{-s}$ is finite and the net force $\sum_{y\in X\setminus\{x\}}\operatorname{sgn}(y-x)\,|y-x|^{-s}$ is zero. Then $X$ is an arithmetic progression. No a priori assumption on the gaps is needed. Subtracting the equilibrium equations of two consecutive points shows that the gaps $g_n$ form a positive harmonic function for an explicit reversible random walk on $\mathbb{Z}$ with long-range jumps. Equilibrium also bounds the ratio of consecutive gaps, by $1.5386\ldots$ when $s=2$. With this bound, an energy estimate shows that the Doob transform of the walk by $g$ is recurrent. Since $1/g$ is a positive harmonic function of the transformed walk, it is constant.

We study domination of the bi-parameter Hilbert transform by $(1,1)$-type sparse forms over axis-parallel rectangles, and show that such domination fails even for smooth test functions compactly supported in $(0,1)^2$. Our proof combines an explicit construction of two finite lattices based on the arithmetic of $\mathbb Z[\sqrt2]$ with a randomization argument.

We prove lower bounds for Fourier transforms of surface measures $$|\widehat{\psi\,d\sigma}(\xi)| \gtrsim_{\psi, K, \sigma, R_0} \sigma(B(p, |\xi|^{-1})), \text{ given }p\in K\Subset\{\psi>0\} ,\ \vec{n}_{p}\parallel \xi,\ |\xi|\ge R_0,$$ on convex hypersurfaces of finite type with at most one vanishing principal curvature at each point. These bounds are uniform in the choice of $p\in K\Subset \{\psi>0\}$ and match the classical upper bound of Bruna-Nagel-Wainger. The key new analytic ingredient is a uniform lower bound for parameter-dependent oscillatory integrals on $\mathbb{R}$ with convex phases. We also give convex examples of finite type with two vanishing principal curvatures for which $$\inf_{p\in K}|\widehat{\psi\,d\sigma}(R \vec{n}_p)|\lesssim_N R^{-N},\ \text{ for arbitrary fixed } \{0\}\Subset K\Subset\{\psi>0\}.$$ Though the estimate above has its own interest, our original motivation is to study Fourier frames for surface measures. In particular, we show that, if $\Gamma$ is a closed convex smooth surface in $\mathbb{R}^d, d\geq 2$, of finite type and at most one principal curvature vanishes at each point, then the surface measure on $\Gamma$ does not admit any Fourier frame. This weakens the non-vanishing curvature condition in a previous result of Iosevich, Lai, the second author, and Wyman. On centrally symmetric surfaces of non-vanishing Gaussian curvature, we obtain more domains without Fourier frames. For example, we show that $\{x\in S^{d-1}:0<x_d<\delta\}, d\geq 3$, does not admit any Fourier frame for all $\delta>0$, which generalizes the result of Chen and the second author on the hemisphere.

We prove ten conjectures of Z.-W. Sun on Ramanujan-type series for $1/\pi^4$ and $\pi^4$ with harmonic numbers. They follow by comparing Taylor coefficients in new Borwein-Bradley-type identities, whose products depend only on $x^3$, $x^4$, $x^6$ or $x^8$. These identities come from parametric extensions of the series of Cullen and Zhao. We derive the extensions from K. C. Au's Wilf-Zeilberger (WZ) seeds, and their sums are trigonometric. Our proofs use Guillera's periodicity argument and give new proofs of the original series. In the same way, Dougall's $_5F_4$ sum gives half-integer analogues of the identities of Koecher, Borwein-Bradley and Cohen, whose Taylor coefficients are Apéry-like series for products of zeta values.

The primary objective of this paper is to establish the pointwise almost everywhere convergence of the singular integrals of Christ-Journé type. To this end, we prove the weak type $(1,1)$ estimate for the corresponding maximal operator defined as \[ T^*[a] f(x):= \sup_{r>0}\left|\int_{|x-y| > r} K(x-y) \int_0^1 a(tx + (1-t)y)\, dt f(y)\, dy\right|, \] where $K$ is a Calderón–Zygmund kernel and $a\in L^\infty$. This result resolves a problem posed by Seeger (Rev. Mat. Iberoam., 2014).

In this article, we study a weighted analogue of the $k$-plane transform, where the weight on the $k$-dimensional measure is given by a radial power of distance. Particularly, we first consider the existence of such operators on Lebesgue spaces imbibed with radial power weights. Then, we proceed to see the mapping properties of the said transform when it acts on radial functions. In this regard, we prove certain weighted $L^p$-improving boundedness and end-point Lorentz space estimates. Finally, we look at the weighted $L^p$-$L^p$ boundedness of the operator acting on general functions, and evaluate its operator norm.

In this contribution we consider the singular periodic problem \[ u''+k\,u=g(u)+e(t),\quad u(0)=u(2\pi),\,\,u'(0)=u'(2\pi), \] where $k\in\R,$ $e:\J\to\R$ is Lebesgue integrable and $g:\I\mapsto\R$ is continuous, but can have a singularity at $x=0.$ In particular, I wish to revisit our earlier result regarding the existence of at least two its solutions and present a proof that is somewhat more transparent than the original one.

We introduce lower and upper macroscopic intermediate dimensions of unbounded subsets of Euclidean space by combining covering contributions from all dyadic annuli beyond a given scale. For every unbounded Borel set $E\subset\R^d$ and $m\in \{1,\ldots,d-1\}$, we prove that the macroscopic intermediate dimensions of $P_VE$ equal the corresponding intrinsic capacity profiles for almost every $m$-dimensional linear subspaces $V\subset \R^d$. One full-measure set works for both dimensions and every parameter $\theta\in[0,1]$. The proof combines a covering–capacity comparison with an exponent gap, fractional-moment estimates for projected coverings, and capacity-weighted retention of source shells. The dimensions recover macroscopic Hausdorff dimension at parameter zero, are invariant under quasi-isometries, and are continuous at positive parameters. The upper macroscopic intermediate dimension agrees with the upper discrete intermediate dimension introduced in \cite{LXZ}, whereas the lower one can differ. We compute the projected dimensions of bounded-base discrete digit Moran sets, including the self-similar case obtained when the bases and digit sets are constant, and give applications to Lipschitz graphs and arithmetic sumsets.

Let $N\in\mathbb N$, $p\in[1,\infty)$, and $Q_0:=(0,1)^N$. In this article, we establish sharp Poincaré inequalities for the nonlocal nonconvex functionals $$ \Phi_{\lambda,p}^\gamma(g;Q_0) :=\lambda^p\iint_{\{(x,y)\in Q_0^2:\,x\ne y,\,|g(x)-g(y)|>\lambda|x-y|^{1+\frac{\gamma}{p}}\}} |x-y|^{\gamma-N}\,dx\,dy, $$ where $g:Q_0\to\mathbb R$ is measurable, $\lambda\in(0,\infty)$, and $\gamma\in\mathbb R$. Specifically, we prove that the estimate $$ \int_{Q_0}\int_{Q_0}|g(x)-g(y)|^p\,dx\,dy \lesssim\lambda^p+\Phi_{\lambda,p}^\gamma(g;Q_0) $$ holds for any such $g$ and $\lambda$ if and only if $\gamma\in(-\infty,-1]$, where the implicit positive constant depends only on $N$, $p$, and $\gamma$. This extends Nguyen's Poincaré inequality for $\gamma=-p$ and provides a useful tool for studying regularity and compactness through these functionals. As applications, we obtain a lower bound for the $\Gamma$-convergence of $\Phi^\gamma_{\lambda,p}$ and related asymptotic characterizations for Sobolev and BV spaces. Moreover, we identify the optimal range of $\gamma$ for which both the Rellich–Kondrachov theorem and a criterion for VMO regularity involving $\Phi^\gamma_{\lambda,p}$ at a fixed threshold hold. These results also answer the question on nonlocal Poincaré inequalities posed by H.-M. Nguyen in [C. R. Math. Acad. Sci. Paris 363 (2025)]. For $\gamma\in(-\infty,-1]$, they further answer questions on Sobolev and BV characterizations posed in the same article and by H. Brezis et al. in [Anal. PDE 17 (2024)].

In his highly influential paper, On the maximal ergodic theorem for certain subsets of the integers, Bourgain introduced the study of pointwise convergence of ergodic averages along randomly generated subsets of the integers, developing a robust \(L^p\)-theory for \(p>1\). More recently, this theory was partially complemented at the \(L^1\) endpoint by work of LaVictoire, who treated averages along random sequences that are “slightly denser" than the set of squares, for which universal \(L^1\) pointwise convergence is known to fail. In this work, we prove that, almost surely, for every measure-preserving system and every integrable function, the ergodic averages formed by sampling along random sequences with quadratic growth converge almost everywhere. Our proof combines harmonic-analytic methods developed by M.~Christ with a “random sampling" of quadratic forms, thereby overcoming the main uniformity obstruction.

We prove an asymmetric version of the discretised ring theorem.

We study finite and countable point configurations in the trace of Brownian motion in $\mathbb{R}^d$, $d\geq2$. In the plane, the set of translations placing a prescribed finite configuration on the trace has Hausdorff dimension two in every nonempty open set; we prove analogous dimension-preservation results for radial and bilipschitz patterns, and every prescribed countable configuration has a dense translation set. For $d\geq3$ we prove a sharp occurrence criterion for regular scale-invariant families of configurations, in terms of the ranks of all labelled subconfigurations. It gives exact thresholds for arithmetic progressions and for homothetic and similar copies of finite sets. In particular we recover and extend a result of Benjamini and Kozma: almost surely the longest arithmetic progression in the trace has length five in $\mathbb{R}^3$, three in $\mathbb{R}^4$ and $\mathbb{R}^5$, and two in $\mathbb{R}^d$ for $d\geq6$.

This is a very first draft of a manuscript about existence of solutions of differential inclusions involving chattering.

We establish sharp bounds for rational points near compact smooth nondegenerate curves in $\mathbb R^n$, for every $n\geq4$. For a nondegenerate curve $\mathcal C$ in $\mathbb R^n$, let \[ \mathcal R_{\mathcal C}(\delta,Q) :=\#\left\{(\mathbf p,q)\in\mathbb Z^n\times\mathbb N: 1\leq q\leq Q,\ \operatorname{dist}(\mathbf p/q,\mathcal C)\leq\delta/q\right\}.\] We prove that \[\mathcal R_{\mathcal C}(\delta,Q) \lesssim_{\varepsilon,\mathcal C} \delta^{n-1}Q^2+Q^\varepsilon \left(Q+\sum_{k=1}^{n-1} \delta^{k^2/(k+1)}Q^{(2k+1)/(k+1)}\right).\] A key new ingredient is a wave packet method developed by Gan–Maldague–Oh.

We study weighted bilinear Hardy-Steklov operators on the half-line for $1<p_1,p_2<\infty$ and $0<q<\infty$. The operator-theoretic core is an exact output-localisation principle. For a bounded bilinear map into $L^q$, compact middle truncations identify compactness with vanishing of a directed output-tail invariant and, with constant one, identify that invariant with the essential distance and a radius-based Hausdorff measure of noncompactness. Under the additional hypothesis of local finite-rank approximability, the same invariant also equals the finite-rank distance and governs the rank-approximation conclusions. We verify these hypotheses for the Hardy-Steklov map and then translate the tail invariant through the five weighted boundedness regions. In Regions $I,\mathrm{II}_1,\mathrm{II}_2,\mathrm{III}_1$, boundedness must be supplemented by lower- and upper-endpoint vanishing, and the essential distance is comparable to the larger endpoint defect. In the strict subcritical Region $\mathrm{III}_2$, boundedness alone implies compactness. This yields a sharp structural transition between the equality boundary $P=Q$ and the summable regime $P>Q$. The quasi-Banach range $0<q<1$ is treated throughout with the powered metric $d_q(f,g)=\|f-g\|_q^q$.

In their seminal work, F. Ricci and E. M. Stein [Proc. Nat. Acad. Sci. USA 83, 1–3 (1986)] introduced the oscillatory singular integral \[ T_Pf(x)=\mathrm{p.v.}\int_{\mathbb{R}^n} e^{iP(x,y)}\frac{\Omega(x-y)}{|x-y|^n}f(y)\,dy, \] and S. Chanillo and M. Christ [Duke Math. J. 55, 141–155 (1987)] established its weak-type $(1,1)$ inequality for smooth $\Omega$. For the rough variant, however, the weak-type $(1,1)$ inequality remained a long-standing open question. In the present paper, we settle this problem: specifically, we prove that the rough oscillatory singular integral is of weak type $(1,1)$ with $\Omega\in L\log^+L(\mathbb{S}^{n-1})$. Moreover, we establish weak-type $(1,1)$ bounds for the rough maximal oscillatory operator associated with $T_P$, a result of independent interest.

We study compactness of the product of two weighted Hardy primitives into $L^q(u)$, $0<q<\infty$, on the input boundary $1\le p_1,p_2\le\infty$, where at least one input exponent is $1$ or $\infty$. For $L^1\times L^p$, $1<p<\infty$, boundedness implies compactness when $q<p/(p+1)$. For $L^1\times L^1$, the corresponding range is $q<1/2$. At and above these boundaries compactness is characterized by additional active-tail decay, and critical power weights give bounded noncompact examples. The endpoint reductions use cumulative Stieltjes measures with their initial atoms and strict Hardy-Copson brackets. An output-mass allocation argument proves local finite-rank density, including quasi-Banach targets. Applying the localization principles developed for the interior-input regime identifies the essential distance, finite-rank distance and Hausdorff noncompactness radius with the coupled active tail, with the appropriate metric power when $q<1$. For a finite left endpoint, ordinary $L^\infty$ inputs admit an exact scalar transfer completed by linear nodal approximation.

We prove a variant of the $L^2$ Fourier restriction inequality for non-degenerate curves in the setting of orthonormal systems. To achieve this, we develop a Hilbert space approach to studying the $L^2$ theory for Fourier restriction for curves. Our arguments also rely on novel $L^p$-improving bounds for associated convolution operators, which take averages over neighbourhoods of cones generated by the curve. The $L^p$-improving bound are established using a variant of the high-low method, in particular using induction-on-scale.

We survey maximal directional averages and directional singular integrals, and their connections with Kakeya–Nikodym phenomena, differentiation theory and the Carleson operator. The exposition is organized by the structure of the set of directions: lacunary and arbitrary finite sets, vector fields as in the conjectures of Zygmund and Stein, and, in higher dimensions, algebraic sets of directions and subspaces. The unifying tool is almost orthogonality, which rests on the ordering of the circle in the plane and on polynomial partitioning in higher dimensions. We explain where it yields sharp or nearly sharp bounds and where new ideas are needed, notably for multiscale averages along algebraic curves and for vector-valued single-annulus estimates, and we collect open problems.

Let $\delta \in [0,1/2]$ and $Q\geq 1$. Given a compact $C^{\infty}$-curve $\mathcal{C}$ in $\mathbb{R}^n$, denote by $N_{\mathcal{C}}(\delta, Q)$ the number of pairs $(\mathbf{a}, q)\in \mathbb{Z}^n\times [Q/2,Q]$ such that the rational point $\mathbf{a}/q$ is $\delta/q$-close to $\mathcal{C}$. We investigate the range of $\delta$ in terms of $Q$ which is necessary for the heuristic $N_{\mathcal{C}}(\delta, Q)\asymp\delta^{n-1}Q^2$ to be correct for the moment curve. For a nondegenerate curve $\mathcal{C}\subset \mathbb{R}^n$ and for sufficiently large $Q$, Hickman and the second author previously established that $N_{\mathcal{C}}(\delta, Q)\lesssim_{\nu} \delta^{n-1}Q^2$ for all $\delta\in[Q^{\alpha(n+1)+\nu},1/2)$ and any $\nu>0$. Here $\alpha(n)=-4n^{-2}+O(n^{-3})$ was explicitly computed. We show that this range is surprisingly close to being sharp in the asymptotic sense as $n\to\infty$, and can at most be extended to $\delta\in[Q^{\alpha(n)},1/2)$. In particular, the sharp exponent can only be about $O(n^{-3})$ better, even though it was widely believed before that an improvement of $O(n^{-1})$ should be possible. The complementary upper bound in this remaining range has been recently established by Gan–Guo–Oh.

The aim of the paper is to provide an extension of the averaging method to the periodic problem for quasilinear differential equations $(\phi(t,u'))' = \varepsilon f(t,u,u',\varepsilon)$, $u(0) = u(T)$, $u'(0) = u'(T)$, where $u$ and $u'$ take values in a Banach space $X$ and $\phi$ is a suitable homeomorphism between a neighborhood of $0$ in $X$ and $X$. The approach follows from a given extension of the averaging method to first order systems of the form $u' = g(t,v)$, $v' = h(t,u,v)$, $u(0) = u(T)$, $v(0) = v(T)$, in a product of Banach spaces, obtained from a direct application of the implicit function theorem in Banach spaces to an equivalent problem. We prove that every nondegenerate zero of the corresponding averaged equation generates, for sufficiently small nonzero values of the parameter, a unique nearby periodic solution. An application is given to an infinite system of coupled forced pendulums with relativistic acceleration.

We develop an abstract averaging method for equations $Mx=\varepsilon N(x,\varepsilon)$ in Banach spaces. The method applies when the equation can be separated into an averaged compatibility condition and a nonlinear complementary problem, and it allows the principal operator $M$ to be genuinely nonlinear. We prove that every nondegenerate zero of the corresponding averaged equation generates, for all sufficiently small nonzero values of $\varepsilon$, a nearby solution. Under a suitable small Lipschitz condition this solution is locally unique, while in the differentiable case it belongs to a unique local $C^1$ branch. The framework contains the abstract linear Fredholm averaging theory as a special case, but also covers quasilinear problems for which the principal operator has no invertible linearization at the reference state. We also determine the leading asymptotic profile of the solutions. Applications are given to a vector $p$-Laplacian Duffing system and to a nonsmooth problem in a Hilbert space.

We develop a measure-theoretic framework for small-order limits of nonlocal energies with metric-valued maps. For globally $L^p$ maps, we consider interaction measures whose marginals have uniformly bounded $L^\infty$ densities $a_\alpha$ and $b_\alpha$. If these densities converge weakly-* to $a_\infty$ and $b_\infty$, the normalized energies converge to the $L^p$ energy weighted by $a_\infty+b_\infty$ if and only if the interaction measures escape every bounded rectangle. For maps that are only locally $L^p$, we introduce target laws describing the distribution of the values sampled at infinity. Convergence of their $p$-distance profiles yields the limiting interaction energy for every $1\leq p<\infty$ and arbitrary Polish targets, with integrated $p$-Wasserstein convergence as a sufficient criterion. The arguments include the endpoint $p=1$ and require no linear structure on the target space. Applications include directional and anisotropic Maz'ya–Shaposhnikova formulas, a Bernoulli-law interpretation of fractional perimeters, and an Abelian principle for heat-semigroup energies.

Let $p$ be an odd prime, let $m\geq1$ be an integer, and let $Q$ be a nondegenerate quadratic form on $\mathbb F_p^{2m}$. For a nonempty set $E\subseteq\mathbb F_p^{2m}$, write $\Delta_Q(E)=\{Q(x-y):x,y\in E\}$. We prove that, whenever $|E|>p^m$, \[ |\Delta_Q(E)|\gg \min\left\{\frac{p}{L_E},\, \frac{|E|}{p^{m-1}L_E^2}\right\}, \qquad L_E=\log\left(2+\frac{p^{m+1}}{|E|}\right), \] with an absolute implied constant independent of $p$, $m$ and $Q$. This confirms the Erdős–Falconer distance conjecture up to a logarithmic factor. In the anisotropic planar case $m=1$, for small sets, we also prove that \[ |\Delta_Q(E)|\gg\frac{|E|}{\log(2|E|)}, \qquad (1\leq |E|\leq p), \] which is also optimal up to a logarithmic factor.