arXiv subjects
Publications and source records attributed to Siddharth Iyer.
We study a problem of Douglass and Ono concerning the smallest integer $n$ such that the partition function $p(n)$ begins with a specified string of digits $f$ in base $b$. By employing an elementary discrepancy framework, we establish new upper bounds that significantly improve upon previous results of Luca.
We establish a lower bound for the frequency with which an irreducible monic cubic polynomial with negative discriminant can be expressed as a sum of two squares ($\square_{2}$). This provides a quantitative answer to a question posed by Grechuk (2021) concerning the infinitude of such values. Our proof relies on a two-dimensional unit argument and the arithmetic of degree six number fields. For example, we show that if $h \equiv 2 \pmod{4}$, then \begin{align*} \# \{n : n^3+h \in \square_{2}, \ 1 \leq n \leq x \} \gg x^{1/3-o(1)}. \end{align*} These arguments may be generalised to study the representation of irreducible monic cubic polynomials by the quadratic form $x^2+ny^2$, where $n \in \mathbb{N}$.
Let $\triangle$ denote the integers represented by the quadratic form $x^2+xy+y^2$ and $\square_{2}$ denote the numbers represented as a sum of two squares. For a non-zero integer $a$, let $S(\triangle,\square_{2},a)$ be the set of integers $n$ such that $n \in \triangle$, and $n + a \in \square_{2}$. We conduct a census of $S(\triangle,\square_{2},a)$ in short intervals by showing that there exists a constant $H_{a} > 0$ with \begin{align*} \# S(\triangle,\square_{2},a)\cap [x,x+H_{a}\cdot x^{5/6}\cdot \log^{19}x] \geq x^{5/6-\varepsilon} \end{align*} for large $x$. To derive this result and its generalization, we utilize a theorem of Tolev (2012) on sums of two squares in arithmetic progressions and analyse the behavior of a multiplicative function found in Blomer, Br{\"u}dern \& Dietmann (2009). Our work extends a classical result of Estermann (1932) and builds upon work of M{\"u}ller (1989).
We prove a sensitivity-to-communication lifting theorem for arbitrary gadgets. Given functions $f: \{0,1\}^n\to \{0,1\}$ and $g : \mathcal X\times \mathcal Y\to \{0,1\}$, denote $f\circ g(x,y) := f(g(x_1,y_1),\ldots,g(x_n,y_n))$. We show that for any $f$ with sensitivity $s$ and any $g$, \[D(f\circ g) \geq s\cdot \bigg(\frac{Ω(D(g))}{\log\mathsf{rk}(g)} - \log\mathsf{rk}(g)\bigg),\] where $D(\cdot)$ denotes the deterministic communication complexity and $\mathsf{rk}(g)$ is the rank of the matrix associated with $g$. As a corollary, we get that if $D(g)$ is a sufficiently large constant, $D(f\circ g) = Ω(\min\{s,d\}\cdot \sqrt{D(g)})$, where $s$ and $d$ denote the sensitivity and degree of $f$. In particular, computing the OR of $n$ copies of $g$ requires $Ω(n\cdot\sqrt{D(g)})$ bits.
We refine a remark of Steinerberger (2024), proving that for $α\in \mathbb{R}$, there exists integers $1 \leq b_{1}, \ldots, b_{k} \leq n$ such that \[ \left\| \sum_{j=1}^k \sqrt{b_j} - α\right\| = O(n^{-γ_k}), \] where $γ_{k} \geq (k-1)/4$, $γ_2 = 1$, and $γ_k = k/2$ for $k = 2^m - 1$. We extend this to higher-order roots. Building on the Bambah-Chowla theorem, we study gaps in $\{x^θ+y^θ: x,y\in \mathbb{N}\cup\{0\}\}$, yielding a modulo one result with $γ_2 = 1$ and bounded gaps for $θ= 3/2$. Given $ρ(m) \geq 0$ with $\sum_{m=1}^{\infty} ρ(m)/m < \infty$, we show that the number of solutions to \[ \left|\sum_{j=1}^{k} a_j^θ - b\right| \leq \frac{ρ\left(\|(a_1, \dots, a_k)\|_{\infty}\right)}{\|(a_1, \dots, a_k)\|_{\infty}^{k}}, \] in the variables $((a_{j})_{j=1}^{k},b) \in \mathbb{N}^{k+1}$ is finite for almost all $θ>0$. We also identify exceptional values of $θ$, resolving a question of Dubickas (2024), by proving the existence of a transcendental $τ$ for which $\|n^τ\| \leq n^v$ has infinitely many solutions for any $v \in \mathbb{R}$.
We define the $\textit{marginal information}$ of a communication protocol, and use it to prove XOR lemmas for communication complexity. We show that if every $C$-bit protocol has bounded advantage for computing a Boolean function $f$, then every $\tilde Ω(C \sqrt{n})$-bit protocol has advantage $\exp(-Ω(n))$ for computing the $n$-fold xor $f^{\oplus n}$. We prove exponentially small bounds in the average case setting, and near optimal bounds for product distributions and for bounded-round protocols.
We prove a lower bound on the communication complexity of computing the $n$-fold xor of an arbitrary function $f$, in terms of the communication complexity and rank of $f$. We prove that $D(f^{\oplus n}) \geq n \cdot \Big(\frac{Ω(D(f))}{\log \mathsf{rk}(f)} -\log \mathsf{rk}(f)\Big )$, where here $D(f), D(f^{\oplus n})$ represent the deterministic communication complexity, and $\mathsf{rk}(f)$ is the rank of $f$. Our methods involve a new way to use information theory to reason about deterministic communication complexity.
We improve upon a result of Steinerberger (2024) by demonstrating that for any fixed $k \in \mathbb{N}$ and sufficiently large $n$, there exist integers $1 \leq a_1, \dots, a_k \leq n$ satisfying: \begin{align*} 0 < \left\| \sum_{j=1}^{k} \sqrt{a_j} \right\| = O(n^{-k/2}). \end{align*} The exponent $k/2$ improves upon the previous exponent of $c k^{1/3}$ of Steinerberger (2024), where $c>0$ is an absolute constant. We also show that for $α\in \mathbb{R}$, there exist integers $1 \leq b_1, \dots, b_k \leq n$ such that: \begin{align*} \left\| \sum_{j=1}^k \sqrt{b_j} - α\right\| = O(n^{-γ_k}), \end{align*} where $γ_k \geq \frac{k-1}{4}$ and $γ_k = k/2$ when $k=2^m - 1$, $m=1,2,\dots$. Importantly, our approach avoids the use of exponential sums.
We show the existence of ``good'' approximations to a real number $γ$ using rationals with denominators formed by digits $0$ and $1$ in base $b$. We derive an elementary estimate and enhance this result by managing exponential sums.
We obtain nontrivial bounds for character sums with multiplicative and additive characters over finite fields over elements with restricted coordinate expansion. In particular, we obtain a nontrivial estimate for such a sum over a finite field analogue of the Cantor set.
Given a function $f$ on $\mathbb{F}_2^n$, we study the following problem. What is the largest affine subspace $\mathcal{U}$ such that when restricted to $\mathcal{U}$, all the non-trivial Fourier coefficients of $f$ are very small? For the natural class of bounded Fourier degree $d$ functions $f:\mathbb{F}_2^n \to [-1,1]$, we show that there exists an affine subspace of dimension at least $ \tildeΩ(n^{1/d!}k^{-2})$, wherein all of $f$'s nontrivial Fourier coefficients become smaller than $ 2^{-k}$. To complement this result, we show the existence of degree $d$ functions with coefficients larger than $2^{-d\log n}$ when restricted to any affine subspace of dimension larger than $Ω(dn^{1/(d-1)})$. In addition, we give explicit examples of functions with analogous but weaker properties. Along the way, we provide multiple characterizations of the Fourier coefficients of functions restricted to subspaces of $\mathbb{F}_2^n$ that may be useful in other contexts. Finally, we highlight applications and connections of our results to parity kill number and affine dispersers.
T2-Shuffling reconstructs multiple sharp T2-weighted images from a single volumetric fast spin-echo (3D-FSE) scan. Wave-CAIPI is a parallel imaging technique that achieves good reconstruction at high accelerations through additional sinusoidal gradients that induce a voxel spreading effect in the readout direction to better take advantage of coil-sensitivity information. In this work, the Shuffling model in T2-Shuffling is augmented with wave-encoding to achieve higher acceleration capability. The resulting "Wave-Shuffling" approach is applied to 3D-FSE and Magnetization-Prepared Rapid Gradient-Echo (MPRAGE) to achieve rapid, 1 mm-isotropic resolution, time-resolved structural imaging.
We give tight bounds on the degree $\ell$ homogenous parts $f_\ell$ of a bounded function $f$ on the cube. We show that if $f: \{\pm 1\}^n \rightarrow [-1,1]$ has degree $d$, then $\| f_\ell \|_\infty$ is bounded by $d^\ell/\ell!$, and $\| \hat{f}_\ell \|_1$ is bounded by $d^\ell e^{\binom{\ell+1}{2}} n^{\frac{\ell-1}{2}}$. We describe applications to pseudorandomness and learning theory. We use similar methods to generalize the classical Pisier's inequality from convex analysis. Our analysis involves properties of real-rooted polynomials that may be useful elsewhere.
Pisier's inequality is central in the study of normed spaces and has important applications in geometry. We provide an elementary proof of this inequality, which avoids some non-constructive steps from previous proofs. Our goal is to make the inequality and its proof more accessible, because we think they will find additional applications. We demonstrate this with a new type of restriction on the Fourier spectrum of bounded functions on the discrete cube.
Purpose: Parallel imaging methods in MRI have resulted in faster acquisition times and improved noise performance. ESPIRiT is one such technique that estimates coil sensitivity maps from the auto-calibration region using an eigenvalue-based method. This method requires choosing several parameters for the the map estimation. Even though ESPIRiT is fairly robust to these parameter choices, occasionally, poor selection can result in reduced performance. The purpose of this work is to automatically select parameters in ESPIRiT for more robust and consistent performance across a variety of exams. Theory and Methods: Stein's unbiased risk estimate (SURE) is a method of calculating an unbiased estimate of the mean squared error of an estimator under certain assumptions. We show that this can be used to estimate the performance of ESPIRiT. We derive and demonstrate the use of SURE to optimize ESPIRiT parameter selection. Results: Simulations show SURE to be an accurate estimator of the mean squared error. SURE is then used to optimize ESPIRiT parameters to yield maps that are optimal in a denoising/data-consistency sense. This improves g-factor performance without causing undesirable attenuation. In-vivo experiments verify the reliability of this method. Conclusion: Simulation experiments demonstrate that SURE is an accurate estimate of expected mean squared error. Using SURE to determine ESPIRiT parameters allows for automatic parameter selections.In-vivo results are consistent with simulation and theoretical results.
We consider distributed computations between two parties carried out over a noisy channel that may erase messages. Following a noise model proposed by Dani et al. (2018), the noise level observed by the parties during the computation in our setting is arbitrary and a priori unknown to the parties. We develop interactive coding schemes that adapt to the actual level of noise and correctly execute any two-party computation. Namely, in case the channel erases $T$ transmissions, the coding scheme will take $N+2T$ transmissions using an alphabet of size $4$ (alternatively, using $2N+4T$ transmissions over a binary channel) to correctly simulate any binary protocol that takes $N$ transmissions assuming a noiseless channel. We can further reduce the communication to $N+T$ by relaxing the communication model and allowing parties to remain silent rather than forcing them to communicate in every round of the coding scheme. Our coding schemes are efficient, deterministic, have linear overhead both in their communication and round complexity, and succeed (with probability 1) regardless of the number of erasures $T$.
The well-known $k$-disjoint path problem ($k$-DPP) asks for pairwise vertex-disjoint paths between $k$ specified pairs of vertices $(s_i, t_i)$ in a given graph, if they exist. The decision version of the shortest $k$-DPP asks for the length of the shortest (in terms of total length) such paths. Similarly the search and counting versions ask for one such and the number of such shortest set of paths, respectively. We restrict attention to the shortest $k$-DPP instances on undirected planar graphs where all sources and sinks lie on a single face or on a pair of faces. We provide efficient sequential and parallel algorithms for the search versions of the problem answering one of the main open questions raised by Colin de Verdiere and Schrijver for the general one-face problem. We do so by providing a randomised $NC^2$ algorithm along with an $O(n^ω)$ time randomised sequential algorithm. We also obtain deterministic algorithms with similar resource bounds for the counting and search versions. In contrast, previously, only the sequential complexity of decision and search versions of the "well-ordered" case has been studied. For the one-face case, sequential versions of our routines have better running times for constantly many terminals. In addition, the earlier best known sequential algorithms (e.g. Borradaile et al.) were randomised while ours are also deterministic. The algorithms are based on a bijection between a shortest $k$-tuple of disjoint paths in the given graph and cycle covers in a related digraph. This allows us to non-trivially modify established techniques relating counting cycle covers to the determinant. We further need to do a controlled inclusion-exclusion to produce a polynomial sum of determinants such that all "bad" cycle covers cancel out in the sum allowing us to count "good" cycle covers.