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Let $$\phi(x,y)$$ be a continuous function, smooth away from the diagonal, such that, for some $$\alpha>0$$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^{\phi}f(x)=\int_{\phi(x,y)=t} f(y) \psi(y) d\sigma_{x,t}(y) \end{equation} map $$L^2({\mathbb R}^d) \to H^{\alpha}({\mathbb R}^d)$$ for all $t>0$. Let $$E$$ be a compact subset of $${\mathbb R}^d$$ for some $$d \ge 2$$, and suppose that the Hausdorff dimension of $$E$$ is $$>d-\alpha$$. We show that any tree graph $$T$$ on $k+1$ ($$k \ge 1$$) vertices is realizable in $$E$$, in the sense that there exist distinct $$x^1, x^2, \dots, x^{k+1} \in E$$ and $t>0$ such that the $$\phi$$-distance $$\phi(x^i, x^j)$$ is equal to $$t$$ for all pairs $(i,j)$ corresponding to the edges of the graph $$T$$.
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Small fractional parts of binary forms
Abstract We obtain bounds on fractional parts of binary forms of the shape $$\Psi(x,y)=\alpha_k x^k+\alpha_l x^ly^{k-l}+\alpha_{l-1}x^{l-1}y^{k-l+1}+\cdots+\alpha_0 y^k$$ with $$\alpha_k,\alpha_l,\ldots,\alpha_0\in{\mathbb R}$$ and $$l\leq k-2.$$ By exploiting recent progress on Vinogradov’s mean value theorem and earlier work on exponential sums over smooth numbers, we derive estimates superior to those obtained hitherto for the best exponent σ, depending on k and $l,$ such that $$ \min_{\substack{0\leq x,y\leq X\\(x,y)\neq (0,0)}}\|\Psi(x,y)\|\leq X^{-\sigma+\epsilon}. $$
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- Award ID(s):
- 2001549
- PAR ID:
- 10507269
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- The Quarterly Journal of Mathematics
- Volume:
- 74
- Issue:
- 4
- ISSN:
- 0033-5606
- Page Range / eLocation ID:
- 1295 to 1330
- Subject(s) / Keyword(s):
- Exponential sums, Diophantine approximation
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/qmath/haad024
