arXiv subjects
Publications and source records attributed to Aleksandre Saatashvili.
We prove that every $n$-letter word over $k$-letter alphabet contains some word as a subsequence in at least $k^{n/4k(1+o(1))}$ many ways, and that this is sharp as $k\to\infty$. For fixed $k$, we show that the analogous number deviates from $μ_k^n$, for some constant $μ_k$, by a factor of at most $n$.
A subset of the Hamming cube over $n$-letter alphabet is said to be $d$-maximal if its diameter is $d$, and adding any point increases the diameter. Our main result shows that each $d$-maximal set is either of size at most $(n+o(n))^d$ or contains a non-trivial Hamming ball. The bound of $(n+o(n))^d$ is asymptotically tight. Additionally, we give a non-trivial lower bound on the size of any $d$-maximal set and show that the number of essentially different $d$-maximal sets is finite.
A spherical $L$-code, where $L \subseteq [-1,\infty)$, consists of unit vectors in $\mathbb{R}^d$ whose pairwise inner products are contained in $L$. Determining the maximum cardinality $N_L(d)$ of an $L$-code in $\mathbb{R}^d$ is a fundamental question in discrete geometry and has been extensively investigated for various choices of $L$. Our understanding in high dimensions is generally quite poor. Equiangular lines, corresponding to $L = \{-α, α\}$, is a rare and notable solved case. Bukh studied an extension of equiangular lines and showed that $N_L(d) = O_L(d)$ for $L = [-1, -β] \cup \{α\}$ with $α,β> 0$ (we call such $L$-codes "uniacute"), leaving open the question of determining the leading constant factor. Balla, Dräxler, Keevash, and Sudakov proved a "uniform bound" showing $\limsup_{d\to\infty} N_L(d)/d \le 2p$ for $L = [-1, -β] \cup \{α\}$ and $p = \lfloor α/β\rfloor + 1$. For which $(α,β)$ is this uniform bound tight? We completely answer this question. We develop a framework for studying uniacute codes, including a global structure theorem showing that the Gram matrix has an approximate $p$-block structure. We also formulate a notion of "modular codes," which we conjecture to be optimal in high dimensions.
In this paper we study the properties of the Lebesgue constant of the conjugate transforms. For conjugate Fejér means we will find necessary and sufficient condition on $t$ for which the estimation $E\left\vert \widetilde{% σ}_{n}^{\left( t\right) }f\right\vert \lesssim E\left\vert f\right\vert $ holds . We also prove that for dyadic irrational $t$, $L\log L $ is maximal Orlicz space for which the estimation $E\left\vert \widetilde{% σ}_{n}^{\left( t\right) }f\right\vert \lesssim 1+E\left( \left\vert f\right\vert \log ^{+}\left\vert f\right\vert \right) $ is valid.