arXiv subjects

N. Kulkarni

Publications and source records attributed to N. Kulkarni.

4 recordsLinked to original sources

arXiv2026

The Erd\H{o}s similarity conjecture and Rajchman measures

Let $A\subseteq\mathbb{R}$ support a probability measure whose Fourier--Stieltjes transform tends to zero at infinity. We prove that, for every $\varepsilon\in(0,1)$, there is a closed, $1$-periodic, nowhere dense set $E\subseteq\mathbb{R}$ such that \[ m(E\cap I)\ge1-\varepsilon \] for every interval $I$ of length $1$, while $E$ contains no affine copy of $A$. Thus every set supporting a Rajchman measure satisfies the Erd\H{o}s similarity conjecture in a uniform large-set form. The proof combines equidistribution modulo one for large dilates of the measure with a multiscale family of low-density periodic blockers; no quantitative rate of Fourier decay is used. We also refine a classical theorem of Iva\v{s}ev-Musatov, showing that for every Hausdorff gauge $h$ there is an $h$-null compact Rajchman support $K$ satisfying \[ \overline{\dim}_{\mathrm B}^{\log} K=\dim_{\mathrm P}^{\log} K=1. \] The value $1$ is sharp for both dimensions.

math.AP↗

arXiv2026

The Erd\H{o}s Similarity Conjecture for Two-Fold Sumsets with a Geometric Summand

We settle a major case in the two-set regime of the Erd\H{o}s similarity conjecture: the sum of a geometric sequence and an arbitrary infinite set is never measure universal. Here a set $E\subset\R$ is measure universal if every measurable set of positive Lebesgue measure contains an affine copy of $E$. More precisely, if $A\subset\R$ is infinite, $a\neq 0$, and $0<|r|<1$, then neither \[ \{ar^n:n\ge 1\}+A \qquad\text{nor}\qquad \{ar^n:n\ge 1\}-A \] is measure universal. More generally, the same conclusion holds when the geometric sequence is replaced by any set containing a lacunary sequence $(b_n)$ with $-\log b_n=O(n)$. Bourgain proved non-universality for sums of three arbitrary infinite sets, whereas the two-set regime is one of the principal remaining cases. Crucially, our conclusion applies to $\{2^{-n}\}+A$ for every infinite $A$, even though the non-universality of $\{2^{-n}\}$ itself remains open. The arbitrary-summand theorem is the maximal lacunary-density endpoint of a general counting-function trade-off. If $S_1,S_2\subset\R$ contain lacunary subsequences and $I(W),J(W)$ count their terms that are at least $e^{-W}$, then $S_1+S_2$ and $S_1-S_2$ are not measure universal whenever \[ \limsup_{W\to\infty}\frac{I(W)J(W)}{W}=\infty. \] No scale-separation or relative-decay assumption is required. The proof combines a finite-grid implementation of Kolountzakis' criterion with a near-additive-energy estimate controlling the clustering of lacunary cross-sums. A packing-number variant replaces lacunarity on one factor by a quantitative metric-mass condition. In particular, for $\alpha_1,\alpha_2>0$, the stretched-exponential sumset \[ \{2^{-n^{\alpha_1}}\}+\{2^{-n^{\alpha_2}}\} \] is not measure universal whenever $1/\alpha_1+1/\alpha_2>1$; analogous conclusions hold for difference sets.

math.CA↗

arXiv2025

The Fourier Ratio and complexity of signals

We study the Fourier ratio of a signal $f:\mathbb Z_N\to\mathbb C$, \[ \mathrm{FR}(f)\ :=\ \sqrt{N}\,\frac{\|\widehat f\|_{L^1(\mu)}}{\|\widehat f\|_{L^2(\mu)}} \ =\ \frac{\|\widehat f\|_1}{\|\widehat f\|_2}, \] as a simple scalar parameter governing Fourier-side complexity, structure, and learnability. Using the Bourgain--Talagrand theory of random subsets of orthonormal systems, we show that signals concentrated on generic sparse sets necessarily have large Fourier ratio, while small $\mathrm{FR}(f)$ forces $f$ to be well-approximated in both $L^2$ and $L^\infty$ by low-degree trigonometric polynomials. Quantitatively, the class $\{f:\mathrm{FR}(f)\le r\}$ admits degree $O(r^2)$ $L^2$-approximants, which we use to prove that small Fourier ratio implies small algorithmic rate--distortion, a stable refinement of Kolmogorov complexity.

math.CA↗

arXiv2012

The T2K Side Muon Range Detector

The T2K experiment is a long baseline neutrino oscillation experiment aiming to observe the appearance of ν e in a νμ beam. The νμ beam is produced at the Japan Proton Accelerator Research Complex (J-PARC), observed with the 295 km distant Super- Kamiokande Detector and monitored by a suite of near detectors at 280m from the proton target. The near detectors include a magnetized off-axis detector (ND280) which measures the un-oscillated neutrino flux and neutrino cross sections. The present paper describes the outermost component of ND280 which is a side muon range detector (SMRD) composed of scintillation counters with embedded wavelength shifting fibers and Multi-Pixel Photon Counter read-out. The components, performance and response of the SMRD are presented.

physics.ins-det↗