arXiv subjects

Nikita Sidorov

Publications and source records attributed to Nikita Sidorov.

At least 19 recordsLinked to original sources

arXiv2025

AIRwaves at CheckThat! 2025: Retrieving Scientific Sources for Implicit Claims on Social Media with Dual Encoders and Neural Re-Ranking

Linking implicit scientific claims made on social media to their original publications is crucial for evidence-based fact-checking and scholarly discourse, yet it is hindered by lexical sparsity, very short queries, and domain-specific language. Team AIRwaves ranked second in Subtask 4b of the CLEF-2025 CheckThat! Lab with an evidence-retrieval approach that markedly outperforms the competition baseline. The optimized sparse-retrieval baseline(BM25) achieves MRR@5 = 0.5025 on the gold label blind test set. To surpass this baseline, a two-stage retrieval pipeline is introduced: (i) a first stage that uses a dual encoder based on E5-large, fine-tuned using in-batch and mined hard negatives and enhanced through chunked tokenization and rich document metadata; and (ii) a neural re-ranking stage using a SciBERT cross-encoder. Replacing purely lexical matching with neural representations lifts performance to MRR@5 = 0.6174, and the complete pipeline further improves to MRR@5 = 0.6828. The findings demonstrate that coupling dense retrieval with neural re-rankers delivers a powerful and efficient solution for tweet-to-study matching and provides a practical blueprint for future evidence-retrieval pipelines.

cs.IR↗

arXiv2023

Bernoulli convolutions -- 2023

Let $θ\in(1,2)$, and $μ_θ$ be the Bernoulli convolution parametrized by $θ$, that is, the measure corresponding to the distribution of the random variable $\sum_{n=1}^{\infty} a_nθ^{-n}$, where the $a_n$ are i.i.d. with probability of $a_n=0$ equal to $\frac12$. As is well known, $μ_θ$ is either equivalent to the Lebesgue measure on $\text{supp}(μ_θ)$, or singular. Recall that an algebraic integer $>1$ is called Pisot if all its other Galois conjugates are smaller than 1 in modulus. It is known that $μ_θ$ is singular with $\dimμ_θ<1$ if $θ$ is Pisot. An algebraic integer $θ$ greater than 1 is called a Salem number if all its other Galois conjugates are of modulus 1, except $θ^{-1}$. I shall prove that (1) $\dimμ_θ=1$ if $θ$ is an algebraic non-Pisot number. (2) if $θ$ is Salem, then $μ_θ$ is equivalent to the Lebesgue measure on $\text{supp}(μ_θ)$, with an unbounded density in $L^p(\text{supp}(μ_θ))$ for all $p<\infty$. (3) Define \[ β_{θ,x,n}=\#\left\{a_1\dots a_n: \exists a_{n+1}\dots\text{such that\ } x=\sum_{k=1}^{\infty}a_nθ^{-k}\right\}. \] Then \[ \lim_{n\to\infty}\sqrt[n]{β_{θ,x,n}}=θ^{\dimμ_θ}\text{\ for}\ μ_θ-\text{a.e.} x. \] (4) Put \[ \bigcup_{n=1}^\infty\left\{\sum_{k=1}^{n}a_kθ^k\mid a_k\in\{-1,0,1\}\right\}= \{y_0(θ)<y_1(θ)<\cdots\}, \] and \[ \ell(θ)=\liminf_{n\to\infty}(y_{n+1}(θ)-y_n(θ)). \] I shall present a short proof of De-Jun Feng's famous theorem which states that $\ell(θ)=0$ for all non-Pisot $θ$.

math.CA↗

arXiv2023

Complex dimensions for IFS with overlaps

The notion of complex dimension of a one-dimensional Cantor set $C=\bigcap_{n=1}^\infty C_n$ dates back decades. It is defined as the set of poles of the meromorphic $ζ$-function $ζ(s)=\sum_{n=1}^{\infty}d_j^s$, where $\Re s>0$, and $d_j$ is the length of the $j$th interval in $C_n$. Following the trend, I switch from sets to measures, which will allow me to generalize the construction to iterated function schemes that do not necessarily satisfy the Open Set Condition.

math.DS↗

arXiv2023

Computing Garsia Entropy for Bernoulli Convolutions with Algebraic Parameters

We introduce a parameter space containing all algebraic integers $β\in(1,2]$ that are not Pisot or Salem numbers, and a sequence of increasing piecewise continuous function on this parameter space which gives a lower bound for the Garsia entropy of the Bernoulli convolution $ν_β$. This allows us to show that $\mathrm{dim}_\mathrm{H} (ν_β)=1$ for all $β$ with representations in certain open regions of the parameter space.

math.CA↗

arXiv2023

On some properties of Perron numbers

Let $θ$ be a real number, $n\in\mathbb N$, and $D_n(θ)=\left\{\sum_{k=1}^n a_kθ^{k}\mid a_k\in\{0,\dots,\lfloor θ\rfloor\}\right\}. $ Let $θ$ be a Perron number, that is, an algebraic integer $>1$ whose other Galois conjugates are less than $θ$ in absolute value. I shall prove two results: (1) $θ^n\ll\#D_n(θ)\ll\sqrt n θ^n.$ (2) $θ$ is of height $\le \lfloorθ\rfloor$.

math.NT↗

arXiv2022

The Minkowski sum of linear Cantor sets

Let $C$ be the classical middle third Cantor set. It is well known that $C+C = [0,2]$ (Steinhaus, 1917). (Here $+$ denotes the Minkowski sum.) Let $U$ be the set of $z \in [0,2]$ which have a unique representation as $z = x + y$ with $x, y \in C$ (the set of uniqueness). It isn't difficult to show that $\dim_H U = \log(2) / \log(3)$ and $U$ essentially looks like $2C$. Assuming $0,n-1 \in A \subset \{0,1,\dots,n-1\}$, define $C_A = C_{A,n}$ as the linear Cantor set which the attractor of the iterated function system \[ \{ x \mapsto (x + a) / n: a \in A \}. \] We consider various properties of such linear Cantor sets. Our main focus will be on the structure of $C_{A,n}+C_{A,n}$ depending on $n$ and $A$ as well as the properties of the set of uniqueness $U_A$.

math.CA↗

arXiv2021

On a family of Self-Affine IFS whose attractors have a non-fractal top

Let $0< λ< μ<1$ and $λ+μ>1$. In this note we prove that for the vast majority of such parameters the top of the attractor $A_{λ,μ}$ of the IFS $\{(λx,μy), (μx+1-μ, λy+1-λ)\}$ is the graph of a continuous, strictly increasing function. Despite this, for most parameters, $A_{λ, μ}$ has a box dimension strictly greater than 1, showing that the upper boundary is not representative of the complexity of the fractal. Finally, we prove that if $λμ\ge 2^{-1/6}$, then $A_{λ,μ}$ has a non-empty interior.

math.DS↗

arXiv2020

Conjugates of Pisot numbers

In this paper we investigate the Galois conjugates of a Pisot number $q \in (m, m+1)$, $m \geq 1$. In particular, we conjecture that for $q \in (1,2)$ we have $|q'| \geq \frac{\sqrt{5}-1}{2}$ for all conjugates $q'$ of $q$. Further, for $m \geq 3$, we conjecture that for all Pisot numbers $q \in (m, m+1)$ we have $|q'| \geq \frac{m+1-\sqrt{m^2+2m-3}}{2}$. A similar conjecture if made for $m =2$. We conjecture that all of these bounds are tight. We provide partial supporting evidence for this conjecture. This evidence is both of a theoretical and computational nature. Lastly, we connect this conjecture to a result on the dimension of Bernoulli convolutions parameterized by $β$, whose conjugate is the reciprocal of a Pisot number.

math.NT↗

arXiv2019

An infinitely generated self-similar set with positive Lebesgue measure and empty interior

Peres and Solomyak asked the question: Do there exist self-similar sets with positive Lebesgue measure and empty interior? This question was answered in the affirmative by Csörnyei et al. They gave a parameterised family of iterated function systems for which almost all of the corresponding self-similar sets satisfied the required properties. They do not however provide an explicit example. Motivated by a desire to construct an explicit example, we in this paper provide an explicit construction of an infinitely generated self-similar set with positive Lebesgue measure and empty interior.

math.DS↗

arXiv2018

Open maps: small and large holes with unusual properties

Let $X$ be a two-sided subshift on a finite alphabet endowed with a mixing probability measure which is positive on all cylinders in $X$. We show that there exist arbitrarily small finite overlapping union of shifted cylinders which intersect every orbit under the shift map. We also show that for any proper subshift $Y$ of $X$ there exists a finite overlapping unions of shifted cylinders such that its survivor set contains $Y$ (in particular, it can have entropy arbitrarily close to the entropy of $X$). Both results may be seen as somewhat counter-intuitive. Finally, we apply these results to a certain class of hyperbolic algebraic automorphisms of a torus.

math.DS↗

arXiv2018

The baker's map with a convex hole

We consider the baker's map $B$ on the unit square $X$ and an open convex set $H\subset X$ which we regard as a hole. The survivor set $\mathcal J(H)$ is defined as the set of all points in $X$ whose $B$-trajectories are disjoint from $H$. The main purpose of this paper is to study holes $H$ for which $\dim_H \mathcal J(H)=0$ (dimension traps) as well as those for which any periodic trajectory of $B$ intersects $\overline H$ (cycle traps). We show that any $H$ which lies in the interior of $X$ is not a dimension trap. This means that, unlike the doubling map and other one-dimensional examples, we can have $\dim_H \mathcal J(H)>0$ for $H$ whose Lebesgue measure is arbitrarily close to one. Also, we describe holes which are dimension or cycle traps, critical in the sense that if we consider a strictly convex subset, then the corresponding property in question no longer holds. We also determine $δ>0$ such that $\dim_H \mathcal J(H)>0$ for all convex $H$ whose Lebesgue measure is less than $δ$. This paper may be seen as a first extension of our work begun in [3, 4, 6, 7, 13] to higher dimensions.

math.DS↗

arXiv2017

A lower bound for the dimension of Bernoulli convolutions

Let $β\in(1,2)$ and let $H_β$ denote Garsia's entropy for the Bernoulli convolution $μ_β$ associated with $β$. In the present paper we show that $H_β>0.82$ for all $β\in (1, 2)$ and improve this bound for certain ranges. Combined with recent results by Hochman and Breuillard-Varjú, this yields $\dim (μ_β)\ge0.82$ for all $β\in(1,2)$. In addition, we show that if an algebraic $β$ is such that $[\mathbb{Q}(β): \mathbb{Q}(β^k)] = k$ for some $k \geq 2$, then $\dim(μ_β)=1$. Such is, for instance, any root of a Pisot number which is not a Pisot number itself.

math.DS↗

arXiv2016

Multidimensional self-affine sets: non-empty interior and the set of uniqueness

Let $M$ be a $d\times d$ contracting matrix. In this paper we consider the self-affine iterated function system $\{Mv-u, Mv+u\}$, where $u$ is a cyclic vector. Our main result is as follows: if $|\det M|\ge 2^{-1/d}$, then the attractor $A_M$ has non-empty interior. We also consider the set $\mathcal U_M$ of points in $A_M$ which have a unique address. We show that unless $M$ belongs to a very special (non-generic) class, the Hausdorff dimension of $\mathcal U_M$ is positive. For this special class the full description of $\mathcal U_M$ is given as well. This paper continues our work begun in two previous papers.

math.DS↗

arXiv2015

Two-dimensional self-affine sets with interior points, and the set of uniqueness

Let $M$ be a $2\times2$ real matrix with both eigenvalues less than~1 in modulus. Consider two self-affine contraction maps from $\mathbb R^2 \to \mathbb R^2$, \begin{equation*} T_m(v) = M v - u \ \ \mathrm{and}\ \ T_p(v) = M v + u, \end{equation*} where $u\neq0$. We are interested in the properties of the attractor of the iterated function system (IFS) generated by $T_m$ and $T_p$, i.e., the unique non-empty compact set $A$ such that $A = T_m(A) \cup T_p(A)$. Our two main results are as follows: 1. If both eigenvalues of $M$ are between $2^{-1/4}\approx 0.8409$ and $1$ in absolute value, and the IFS is non-degenerate, then $A$ has non-empty interior. 2. For almost all non-degenerate IFS, the set of points which have a unique address is of positive Hausdorff dimension -- with the exceptional cases fully described as well. This paper continues our work begun in [11].

math.DS↗

arXiv2015

On a family of self-affine sets: topology, uniqueness, simultaneous expansions

Let $β_1,β_2>1$ and $T_i(x,y) = \bigl(\frac{x+i}{β_1}, \frac{y+i}{β_2}\bigr),\ i\in\{\pm1\}$. Let $A := A_{β_1, β_2}$ be the unique compact set satisfying $A = T_{1}(A) \cup T_{-1}(A)$. In this paper we give a detailed analysis of $A$, and the parameters $(β_1, β_2)$ where$A$ satisfies various topological properties. In particular, we show that if $β_1<β_2<1.202$,then $A$ has a non-empty interior, thus significantly improving the bound from [1]. In the opposite direction,we prove that the connectedness locus for this family studied in [16] is not simply connected.We prove that the set of points of $A$ which have a unique address has positive Hausdorff dimension for all $(β_1,β_2)$.Finally, we investigate simultaneous $(β_1,β_2)$-expansions of reals, which were the initial motivation for studying this family in [5].

math.DS↗

arXiv2014

Expansions in non-integer bases: lower order revisited

Let $q\in(1,2)$ and $x\in[0,\frac1{q-1}]$. We say that a sequence $(\varepsilon_i)_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ is an expansion of $x$ in base $q$ (or a $q$-expansion) if \[ x=\sum_{i=1}^{\infty}\varepsilon_iq^{-i}. \] For any $k\in\mathbb N$, let $\mathcal B_k$ denote the set of $q$ such that there exists $x$ with exactly $k$ expansions in base $q$. In [12] it was shown that $\min\mathcal B_2=q_2\approx 1.71064$, the appropriate root of $x^{4}=2x^{2}+x+1$. In this paper we show that for any $k\geq 3$, $\min\mathcal B_k=q_f\approx1.75488$, the appropriate root of $x^3=2x^2-x+1$.

math.NT↗

arXiv2014

On cycles for the doubling map which are disjoint from an interval

Let $T:[0,1]\to[0,1]$ be the doubling map and let $0<a<b<1$. We say that an integer $n\ge3$ is bad for $(a,b)$ if all $n$-cycles for $T$ intersect $(a,b)$. Let $B(a,b)$ denote the set of all $n$ which are bad for $(a,b)$. In this paper we completely describe the sets: \[ D_2=\{(a,b) : B(a,b)\,\text{is finite}\} \] and \[ D_3=\{(a,b) : B(a,b)=\varnothing\}. \] In particular, we show that if $b-a<\frac16$, then $(a,b)\in D_2$, and if $b-a\le\frac2{15}$, then $(a,b)\in D_3$, both constants being sharp.

math.DS↗

arXiv2013

Supercritical holes for the doubling map

For a map $S:X\to X$ and an open connected set ($=$ a hole) $H\subset X$ we define $\mathcal J_H(S)$ to be the set of points in $X$ whose $S$-orbit avoids $H$. We say that a hole $H_0$ is supercritical if (i) for any hole $H$ such that $\bar{H_0}\subset H$ the set $\mathcal J_H(S)$ is either empty or contains only fixed points of $S$; (ii) for any hole $H$ such that $\barH\subset H_0$ the Hausdorff dimension of $\mathcal J_H(S)$ is positive. The purpose of this note to completely characterize all supercritical holes for the doubling map $Tx=2x\bmod1$.

math.DS↗