arXiv subjects

Chun-Kai Tseng

Publications and source records attributed to Chun-Kai Tseng.

4 recordsLinked to original sources

arXiv2026

An $m^{2.943}$ Bohnenblust--Hille Bound on the Boolean Cube

Let $q_m=2m/(m+1)$ and put \[ β_0=\frac{3}{2}+\frac{1}{\log 2}=2.9426950408\ldots, \] where $\log$ is the natural logarithm. We give a proof scheme showing that, for every $\varepsilon>0$, there is $C_\varepsilon<\infty$ such that every complex-valued function $f:\{-1,1\}^n\to\C$ of Fourier degree at most $m$ satisfies \[ \left(\sum_{A\subseteq[n]}\abs{\wh f(A)}^{q_m}\right)^{1/q_m} \le C_\varepsilon m^{β_0+\varepsilon}\norm{f}_\infty. \] The improvement over the $m^9$ estimate of the earlier draft has two ingredients. The first three Fourier levels are estimated at the scales $m$, $m^{3/2}$, and $m^{5/3}$. These losses are encoded in the weight $M^{μ_r/r}r^B\min\{r,L_M\}^8$, $L_M\asymp_B \log (M+1)$, with $μ_r=\lceil3r/2\rceil$ for $r\ge2$. A parity-compatible central window handles $r 1/\log2$. The formula for $μ_r$ is explained below.

cs.IT↗

arXiv2026

Small moments of the sensitivity of polynomial threshold functions

In the first version of Chang, Slote, Volberg, and Zhang's paper \cite{BSA_of_PTF}, the authors modify a nice recursive approach due to Kane in \cite{Correct_exponent_for_AS} where he bounded the average sensitivity of polynomial threshold functions. In \cite{BSA_of_PTF} Kane's argument was adopted to estimate the boolean surface area of polynomial threshold function. The bridge is a combinatorial averaging lemma considering all balanced partitions. The lemma serves as a substitute for an additive property of average sensitivity. With the lemma, one can apply a Kane-type algorithm to derive a recurrence. Solving the recurrence then gives an upper bound of $e^{C_d \sqrt{\log n}}$ for the boolean surface area. In the second version of the same paper, the authors derive a polylog upper bound for BSA of PTFs. The difference is that they use a tail estimate for the sensitivity function. With the help of a polynomial restriction lemma in \cite{poly_restriction} they sharpen the upper bound. It is noteworthy that when applying the polynomial restriction, each coordinate is put into each part independently with equal probability. As a result, a partition does not necessarily have equal-size blocks. In other words, it may not be balanced. In this note, we first investigate the effect of different partitioning. Second, we use the recursive method in the first version to derive a polylog upper bound for $\mathbb E[s(x)^η]$ where $η< 1/2$. It is interesting to note the phase transition that happens at $η=1/2$ in both versions of the proof (but in a completely different form). Section 2.5 treats that.

math.PR↗

arXiv2025

Falconer's problem for dot product on paraboloids

We establish dimensional thresholds for dot product sets associated with compact subsets of translated paraboloids. Specifically, we prove that when the dimension of such a subset exceeds $ \frac{5}{4} = \frac{3}{2} - \frac{1}{4} $ in $\mathbb{R}^3$, and $ \frac{d}{2} - \frac{1}{4} - \frac{1}{8d - 4} $ in $\mathbb{R}^d$ for $d\geq 4$, its dot product set has positive Lebesgue measure. This result demonstrates that if a compact set in $ \mathbb{R}^d $ exhibits a paraboloidal structure, then the usual dimensional barrier of $ \frac{d}{2} $ for dot product sets can be lowered for $ d \geq 3 $. Our work serves as the continuous counterpart of a paper by Che-Jui Chang, Ali Mohammadi, Thang Pham, and Chun-Yen Shen, which examines the finite field setting with partial reliance on the extension conjecture. The key idea, closely following their paper, is to reformulate the dot product set on the paraboloid as a variant of a distance set. This reformulation allows us to leverage state-of-the-art results from the pinned distance problem, as established by Larry Guth, Alex Iosevich, Yumeng Ou, and Hong Wang for $ d = 2 $, and by Xiumin Du, Yumeng Ou, Kevin Ren, and Ruixiang Zhang for higher dimensions. Finally, we present explicit constructions and existence proofs that highlight the sharpness of our results.

math.CO↗

arXiv2016

Link Quality Control Mechanism for Selective and Opportunistic AF Relaying in Cooperative ARQs: A MLSD Perspective

Incorporating relaying techniques into Automatic Repeat reQuest (ARQ) mechanisms gives a general impression of diversity and throughput enhancements. Allowing overhearing among multiple relays is also a known approach to increase the number of participating relays in ARQs. However, when opportunistic amplify-and-forward (AF) relaying is applied to cooperative ARQs, the system design becomes nontrivial and even involved. Based on outage analysis, the spatial and temporal diversities are first found sensitive to the received signal qualities of relays, and a link quality control mechanism is then developed to prescreen candidate relays in order to explore the diversity of cooperative ARQs with a selective and opportunistic AF (SOAF) relaying method. According to the analysis, the temporal and spatial diversities can be fully exploited if proper thresholds are set for each hop along the relaying routes. The SOAF relaying method is further examined from a packet delivery viewpoint. By the principle of the maximum likelihood sequence detection (MLSD), sufficient conditions on the link quality are established for the proposed SOAF-relaying-based ARQ scheme to attain its potential diversity order in the packet error rates (PERs) of MLSD. The conditions depend on the minimum codeword distance and the average signal-to-noise ratio (SNR). Furthermore, from a heuristic viewpoint, we also develop a threshold searching algorithm for the proposed SOAF relaying and link quality method to exploit both the diversity and the SNR gains in PER. The effectiveness of the proposed thresholding mechanism is verified via simulations with trellis codes.

cs.IT↗