We obtain nontrivial exponents for Erdős–Falconer type point configuration problems. Let T_k(E) denote the set of distinct congruent k -dimensional simplices determined by (k+1) -tuples points from E . For 1 \le d , we prove that there exists a t_{k,d} < such that, if \subset {\mathbb R}^d \ge 2 with \mathrm {dim}_{{\mathcal H}}(E)>t_{k,d} then {k+1 \choose 2} -imensional Lebesgue measure is positive. Results this were previously obtained triangles in plane (k=d=2) [8] and higher [7]....

For functions F, G on R n , any k-dimensional affine subspace H ⊂ 1 ≤ k < n, and exponents p, q, r ≥ 2 with p +

Abstract We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide general mechanism to Falconer-type problems involving ( k +1)-point configurations in geometric measure theory, with ≥ 2, including distribution of simplices, volumes and angles determined by points fractal subsets E ⊂ ℝ d , 2. If T ) denotes set noncongruent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo>...

Abstract We introduce a class of Falconer distance problems, which we call restricted type, lying between the classical version and its pinned variant. Prototypical sets are diagonal sets, k -point configuration given by $$ \begin{align*}\Delta^{\mathrm{diag}}(E)= \{ \,|(x,x,\dots,x)-(y_1,y_2,\dots,y_{k-1})| : x, y_1, \dots,y_{k-1} \in E\, \}\end{align*} for compact $E\subset \mathbb {R}^d$ $k\ge 3$ . show that $\Delta ^{\mathrm{diag}}(E)$ has non-empty interior if Hausdorff dimension E...

We find the sharp range for boundedness of discrete bilinear spherical maximal function dimensions d ⩾ 5 . That is, we show that this operator is bounded on l p ( Z ) × q → r 1 / + and > − 2 sharp. Our approach mirrors used by Jeong Lee in continuous setting. For = 3 , 4 our previous work, which different techniques, still gives best known bounds. also prove analogous results higher degree k, ℓ-linear operators.

For any dynamical system, we show that higher variation-norms for the sequence of ergodic bilinear averages two functions satisfy a large range Lp estimates. It follows that, with probability one, number fluctuations along this may grow at most polynomially respect to (the growth of) underlying scale. These results strengthen previous works Lacey and Bourgain where almost surely convergence was proved (which is equivalent qualitative statement finite each scale). Via transference, proof...

Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers, where the numbers satisfy $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$, $F_1=1$ and $F_2=2$. The distribution number summands in such decomposition converges to Gaussian, gaps between geometric decay, longest gap is similar run heads biased coin; these results also hold more generally, though technical reasons previous work needed assume coefficients recurrence relation are...

Abstract Zeckendorf's theorem states that every positive integer can be decomposed uniquely as a sum of nonconsecutive Fibonacci numbers. The distribution the number summands converges to Gaussian, and individual measures on gajw between for m € [F n ,F n+1 ) converge geometric decay almost all n→ ∞. While similar results are known many other recurrences, previous work focused proving Gaussianity or average gap measure. We derive general conditions, which easily checked, yield in generalized...

We prove that if the Hausdorff dimension of $E \subset {\Bbb R}^d$, $d \ge 3$, is greater than $\min \left\{ \frac{dk+1}{k+1}, \frac{d+k}{2} \right\},$ then ${k+1 \choose 2}$-dimensional Lebesgue measure $T_k(E)$, set congruence classes $k$-dimensional simplexes with vertices in $E$, positive. This improves best bounds previously known, decreasing $\frac{d+k+1}{2}$ threshold obtained Erdoğan-Hart-Iosevich (2012) to $\frac{d+k}{2}$ via a different and conceptually simpler method. also give...

We obtain nontrivial exponents for Erd\H os-Falconer type problems. Let $T_k(E)$ denote the set of distinct congruent $k$-dimensional simplexes determined by $(k+1)$-tuples points from $E$. prove that there exists $s_0(d)<d$ such that, if $E \subset {\Bbb R}^d,\, d \ge 2$, with $dim_{{\mathcal H}}(E)>s_0(d)$, then ${k+1 \choose 2}$-dimensional Lebesgue measure is positive. Results were previously obtained triangles in plane \cite{GI12} and higher dimensions \cite{GGIP12}. In this paper, we...

We prove variation-norm estimates for the Walsh model of truncated bilinear Hilbert transform, extending related results Lacey, Thiele, and Demeter. The proof uses analysis on phase plane two new ingredients: (i) a variational extension lemma Bourgain by Nazarov–Oberlin–Thiele, (ii) Rademacher–Menshov theorem Lewko–Lewko.

We study a generalization of the Erdős unit distance problem to chains <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> distances. Given alttext="script upper P comma"> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi>...

Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers. The distribution the number summands converges to Gaussian, and individual measures on gaps between for $m \in [F_n, F_{n+1})$ converge geometric decay almost all $m$ $n\to\infty$. While similar results are known many other recurrences, previous work focused proving Gaussianity or average gap measure. We derive general conditions which easily checked yield in...

We prove that if the Hausdorff dimension of a compact subset ${\mathbb R}^d$ is greater than $\frac{d+1}{2}$, then set angles determined by triples points from this has positive Lebesgue measure. Sobolev bounds for bi-linear analogs generalized Radon transforms and method stationary phase play key role. These results complement those V. Harangi, T. Keleti, G. Kiss, P. Maga, Mattila B. Stenner in (\cite{HKKMMS10}). also obtain new upper number times an angle can occur among $N$ R}^d$, $d \ge...