Seungki Kim

University of Cincinnati

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Tsuyoshi Takagi

The University of Tokyo

Woohun Lee

Korea Advanced Institute of Science and Technology

Jíntai Ding

University of Cincinnati

Bo‐Yin Yang

Academia Sinica

Akshay Venkatesh

Stanford University

Mishel Skenderi

University of Utah

Yuntao Wang

Xi'an Jiaotong University

Hyun-Sook Choi

National Science Museum

Phong Q. Nguyễn

École Normale Supérieure - PSL

Yuntao Wang

University of Electro-Communications

Cooperative Institutions

Korea Advanced Institute of Science and Technology

Kootenay Association for Science & Technology

Tsinghua University

University of Pennsylvania

The University of Tokyo

Kyushu University

Osaka University

Nanjing Tech University

Zhongda Hospital Southeast University

Jinshan Hospital of Fudan University

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SOUND BOUND

Seungki Kim Woohun Lee

SOUND BOUND is an interactive graphic equalizer. It displays the magnitude of frequency akin to traditional graphic equalizers or other sound visualizers, and it also allows the user to distort and create sounds by touch interaction. The user can create virtual balls or instrument icons on the screen and can cause these objects to collide with the equalizer by using a physics engine. When virtual balls collide with the equalizer bars, the magnitude of certain frequencies is amplified. When equalizer bars strikes an instruments icon, it create an instrumental sound effect. Until the invention of SOUND BOUND, the activities involved in listening to music have been essentially passive and static. SOUND BOUND actively engages people in the activity of listening to music and enhances their experience as well.

10.1145/2468356.2479585

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Citations (2)

Mean value formulas on sublattices and flags of the random lattice

arXiv (Cornell University) (2020)

Seungki Kim

We present the extensions of the Siegel integral formula ([10]), which counts the vectors of the random lattice, to the context of counting its sublattices and flags. Perhaps surprisingly, it turns out that many quantities of interest diverge to infinity.

Lattice (music)

FLAGS register

Infinity

10.48550/arxiv.2005.10874

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Citations (2)

Higher-rank pointwise discrepancy bounds and logarithm laws for generic lattices

arXiv (Cornell University) (2021)

Seungki Kim Mishel Skenderi

We prove a higher-rank analogue of a well-known result of W. M. Schmidt concerning almost everywhere pointwise discrepancy bounds for lattices in Euclidean space (see Theorem 1 [Trans. Amer. Math. Soc. 95 (1960), 516-529]). We also establish volume estimates pertaining to higher minima of lattices and then use the work of Kleinbock-Margulis and Kelmer-Yu to prove dynamical Borel-Cantelli lemmata and logarithm laws for higher minima and various related functions.

Pointwise

Rank (graph theory)

Maxima and minima

Pointwise convergence

Iterated logarithm

10.48550/arxiv.2107.12510

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Why 1.02? The root Hermite factor of LLL and stochastic sandpile models.

arXiv (Cornell University) (2018)

Jíntai Ding Seungki Kim Tsuyoshi Takagi Yuntao Wang

In lattice-based cryptography, a disturbing and puzzling fact is that there exists such a conspicuous gap between the actual performance of LLL and what could be said of it theoretically. By now, no plausible mathematical explanation is yet proposed. In this paper, we provide compelling evidence that LLL behaves essentially identically to a certain stochastic variant of the sandpile model that we introduce. This allows us to explain many observations on the LLL algorithm that have so far been considered mysterious. For example, we can now present a mathematically well- substantiated explanation as to why LLL has the root Hermite factor (RHF) $\approx$ 1.02 and why the LLL algorithm can not hit the basis with the root Hermite factor (RHF) $\approx$ 1.074, the theoretical upper bound. Our approach also shows strongly that minor modifications of LLL without incurring a significant increase in computational cost to boost its RHF is very unlikely to occur. This should boost very much our confidence on how we can use LLL to help us to select practical parameters for lattice-based cryptosystems, a critical and practical problem we must resolve in order to select the best post-quantum cryptographic standards in the near future.

Root (linguistics)

Lattice (music)

Source

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On counterexamples to the Mertens conjecture

Research in Number Theory (2024)

Seungki Kim Phong Q. Nguyễn

Counterexample

10.1007/s40993-024-00603-9

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Bounds on the gaps in Kronecker sequences (and a little bit more)

arXiv (Cornell University) (2023)

Seungki Kim

We provide bounds on the sizes of the gaps -- defined broadly -- in the set $\{k_1\vbeta_1 + \ldots + k_n\vbeta_n \mbox{ (mod 1)} : k_i \in \Z \cap (0,Q^\frac{1}{n}]\}$ for generic $\vbeta_1, \ldots, \vbeta_n \in \R^m$ and all sufficiently large $Q$. We also introduce a related problem in Diophantine approximation, which we believe is of independent interest.

10.48550/arxiv.2312.02414

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Random lattice vectors in a set of size O(n)

arXiv (Cornell University) (2016)

Seungki Kim

We adopt the sieve ideas of Schmidt and S\"odergren in order to study the statistics of vectors of a random lattice of dimension n contained in a set of volume O(n). We also give some sporadic applications of our results to number theory.

Lattice (music)

Sieve (category theory)

10.48550/arxiv.1611.02880

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Citations (3)

Adelic Rogers integral formula

arXiv (Cornell University) (2022)

Seungki Kim

We formulate and prove the extension of the Rogers integral formula to the adeles of number fields. We also prove the second moment formulas for a few important cases, enabling a number of classical and recent applications of the formula to extend immediately to any number field.

10.48550/arxiv.2205.03138

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On the distribution of lengths of short vectors in a random lattice

Mathematische Zeitschrift (2015)

Seungki Kim

Lattice (music)

Sieve (category theory)

10.1007/s00209-015-1580-y

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Citations (9)

Counting rational points of a Grassmannian