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Dmitry Krachun

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math.NT math-ph math.MP math.PR Data Structures and Algorithms Machine Learning math.MG nlin.SI

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preprint2026arXiv

Provable Quantization with Randomized Hadamard Transform

Vector quantization via random projection followed by scalar quantization is a fundamental primitive in machine learning, with applications ranging from similarity search to federated learning and KV cache compression. While dense random rotations yield clean theoretical guarantees, they require $Θ(d^2)$ time. The randomized Hadamard transform $HD$ reduces this cost to $O(d \log d)$ , but its discrete structure complicates analysis and leads to weaker or purely empirical compression guarantees. In this work, we study a variant of this approach: dithered quantization with a single randomized Hadamard transform. Specifically, the quantizer applies $HD$ to the input vector and subtracts a random scalar offset before quantizing, injecting additional randomness at negligible cost. We prove that this approach is unbiased and provides mean squared error bounds that asymptotically match those achievable with truly random rotation matrices. In particular, we prove that a dithered version of TurboQuant achieves mean squared error $\bigl(π\sqrt{3}/2 + o(1)\bigr) \cdot 4^{-b}$ at $b$ bits per coordinate, where the $o(1)$ term vanishes uniformly over all unit vectors and all dimensions as the number of quantization levels grows.

preprint2022arXiv

On the six-vertex model's free energy

In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime $Δ<1$ . As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches $1/2$ . This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when $a=b=1$ and $c\ge1$ , and the rotational invariance of the six-vertex model and the Fortuin-Kasteleyn percolation.

preprint2020arXiv

On some determinants involving Jacobi symbols

In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer $n\equiv3\pmod4$ , we show that $(6,1)_n=[6,1]_n=(3,2)_n=[3,2]_n=0$ and $(4,2)_n=(8,8)_n=(3,3)_n=(21,112)_n=0$ as conjectured by Sun, where $(c,d)_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{1\le i,j\le n-1}$ and $[c,d]_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{0\le i,j\le n-1}$ with $(\frac{\cdot}n)$ the Jacobi symbol. We also prove that $(10,9)_p=0$ for any prime $p\equiv5\pmod{12}$ , and $[5,5]_p=0$ for any prime $p\equiv 13,17\pmod{20}$ , which were also conjectured by Sun. Our proofs involve character sums over finite fields.

preprint2020arXiv

On sums of four pentagonal numbers with coefficients

The pentagonal numbers are the integers given by $p_5(n)=n(3n-1)/2\ (n=0,1,2,\ldots)$ . Let $(b,c,d)$ be one of the triples $(1,1,2),(1,2,3),(1,2,6)$ and $(2,3,4)$ . We show that each $n=0,1,2,\ldots$ can be written as $w+bx+cy+dz$ with $w,x,y,z$ pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integer is a sum of five pentagonal numbers two of which are equal; this refines a classical result of Cauchy claimed by Fermat.

preprint2020arXiv

Triangles on $\mathbb{Z}^2$ and sliding phenomenon

We confirm the list from \cite{MSS} of values $D$ for which the high-density hard-core model on $\mathbb{Z}^2$ with exceptional distance $D$ has infinitely many extremal Gibbs states. As a byproduct, we prove that for all $D>0$ there exists an acute-angled triangle inscribes in $\mathbb{Z}^2$ with side-lengths at least $D$ and area $\sqrt{3}/4\cdot D^2+O(D^{4/5})$ .

preprint2016arXiv

On random partitions induced by random maps

The lattice of the set partitions of $[n]$ ordered by refinement is studied. Given a map $ϕ: [n] \rightarrow [n]$ , by taking preimages of elements we construct a partition of $[n]$ . Suppose $t$ partitions $p_1,p_2,\dots,p_t$ are chosen independently according to the uniform measure on the set of mappings $[n]\rightarrow [n]$ . The probability that the coarsest refinement of all $p_i$ 's is the finest partitions $\{\{1\},\dots,\{n\}\}$ is shown to approach $1$ for any $t\geq 3$ and $e^{-1/2}$ for $t=2$ . The probability that the finest coarsening of all $p_i$ 's is the one-block partition is shown to approach 1 if $t(n)-\log{n}\rightarrow \infty$ and $0$ if $t(n)-\log{n}\rightarrow -\infty$ . The size of the maximal block of the finest coarsening of all $p_i$ 's for a fixed $t$ is also studied.

preprint2016arXiv

On sums of triangular numbers

We study decompositions of natural numbers into triangular summands. For instance, we prove that any natural number can be represented as a sum of four triangular numbers, two of them having even indices and the other two having odd indices. This settles a conjecture by Zhi-Wei Sun (arXiv:1503.03743v8).

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Dmitry Krachun

Research overview

Is this your work?

7 publications

Provable Quantization with Randomized Hadamard Transform

On the six-vertex model's free energy

On some determinants involving Jacobi symbols

On sums of four pentagonal numbers with coefficients

Triangles on $\mathbb{Z}^2$ and sliding phenomenon

On random partitions induced by random maps

On sums of triangular numbers

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Building this map preview

Dmitry Krachun

Research overview

Is this your work?

7 publications

Provable Quantization with Randomized Hadamard Transform

On the six-vertex model's free energy

On some determinants involving Jacobi symbols

On sums of four pentagonal numbers with coefficients

Triangles on Z2\mathbb{Z}^2Z2 and sliding phenomenon

On random partitions induced by random maps

On sums of triangular numbers

See the researcher in context

Building this map preview

Triangles on $\mathbb{Z}^2$ and sliding phenomenon