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Prasuna Bandi

Publications and source records attributed to Prasuna Bandi.

8 recordsLinked to original sources
arXiv2026

Averaged Fourier Estimates and Dyadic Approximation on the Cantor set

Let $C$ be the middle-third Cantor set and let $\mu$ be the natural Cantor probability measure. Let \[ \gamma=\frac{\log2}{\log3}. \] The two main results of this paper are \[ \mu\{x\in C:\|2^n x\| 2-\gamma. \] and \[ \mu\{x\in C:\|2^n x\|<n^{-\tau}\text{ for infinitely many }n\}=1 \qquad \text{ for } \tau<\frac{1-\gamma}{2}. \] These results give new progress toward Velani's conjecture on zero-one law for dyadic approximation in the middle-third Cantor set.

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arXiv2026

On the Hausdorff Dimension of weighted exactly Approximable Vectors

We show that the Hausdorff dimension of $\boldsymbol w$-weighted $\tau$-exactly approximable vectors in $\mathbb R^d$ coincides with the Hausdorff dimension of $\boldsymbol w$-weighted $\tau$-approximable vectors, generalizing a result of the first named author and De Saxc\'e.

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arXiv2025

Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation

In this paper, we prove a new ergodic theorem for $\mathbb{R}^d$-actions involving averages over dilated submanifolds, thereby generalizing the theory of spherical averages. Our main result is a quantitative estimate for the error term of such averages valid for smooth functions under some effective mixing assumptions on the action. With the aid of this theorem, we investigate multiplicative-type Dirichlet-improvability for $(m\times n)$-matrices with real coefficients. In particular, we establish that almost all matrices are uniformly approximable by the function $x\mapsto x^{-1}(\log x)^{-1+\varepsilon}$ for any $\varepsilon>0$. Results of this type motivate a question which can be thought as a strengthening of Littlewood's conjecture in multiplicative Diophantine approximation.

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arXiv2023

Hausdorff dimension and exact approximation order in $\mathbb{R}^n$

Given a non-increasing function $ψ\colon\mathbb{N}\to\mathbb{R}^+$ such that $s^{\frac{n+1}{n}}ψ(s)$ tends to zero as $s$ goes to infinity, we show that the set of points in $\mathbb{R}^n$ that are exactly $ψ$-approximable is non-empty, and we compute its Hausdorff dimension. For $n\geq 2$, this answers questions of Jarník and of Beresnevich, Dickinson, and Velani.

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arXiv2022

Exact approximation order and well-distributed sets

We prove that for any proper metric space $X$ and a function $ψ:(0,\infty)\to(0,\infty)$ from a suitable class of approximation functions, the Hausdorff dimensions of the set $W_ψ(Q)$ of all points $ψ$-well-approximable by a well-distributed subset $Q\subset X$, and the set $E_ψ(Q)$ of points that are exactly $ψ$-approximable by $Q$, coincide. This answers in a general setting, a question of Beresnevich-Dickinson-Velani in the case of approximation of reals by rationals, and answered by Bugeaud in that case using the continued-fraction expansion of reals. Our main result applies in particular to approximation by orbits of fixed points of a wide class of discrete groups of isometries acting on the boundary of hyperbolic metric spaces.

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arXiv2020

Small solutions of quadratic forms with congruence conditions

We consider a system of homogeneous quadratic forms with congruence conditions in $n\geq 3$ variables and prove the existence of two linearly independent integral solutions of bounded height. We also show the existence of small height integral zeros of this system avoiding a given set of hyperplanes.

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arXiv2020

A generic effective Oppenheim theorem for systems of forms

We prove a uniform effective density theorem as well as an effective counting result for a generic system comprising a polynomial with a mild homogeneous condition and several linear forms using Roger's second moment formula for the Siegel transform on the space of unimodular lattices.

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