We study the distribution of the number of coprime lattice points in randomly translated disks. We prove that its spatial variance is asymptotic to $\sigma^2r\log r$ and establish a central limit theorem under the corresponding normalization, where $\sigma^2$ is an explicit positive constant. By truncating the numerators of rational frequencies in lowest terms, the proof reduces higher-moment estimates to counting squarefree denominators in zero-sum relations.
We establish a uniform two-dimensional local limit theorem for the joint distribution of $\Omega(n)$ and $\Omega(n+1)$. More precisely, along a set of scales of logarithmic density one, the normalized joint counts are uniformly approximated by the density of two independent Gaussian random variables, each with mean and...
Let the center of a $d$-dimensional Euclidean ball be uniformly distributed modulo $\mathbb Z^d$. We study the resulting number of integer lattice points as the dimension and radius tend to infinity. The critical scale is $4\pi^2R_d^2/(d+2)=\log d$. In a logarithmic window around this scale, the logarithm of the lattic...
We give an explicit finite sum for the number of ideals of the Lie algebra of strictly lower triangular $n\times n$ matrices over $\mathbb F_q$, valid for every prime power $q$. The sum runs over integer compositions, with weights expressed using ordinary and Gaussian binomial coefficients. A contraction bijection tran...
For a polynomial of degree $n$ whose zeros lie in the closed unit disk, we determine the largest possible distance from a prescribed critical point of modulus $r$ to the nearest zero. If $n$ is even, the sharp radius is $\sqrt{1-r^2}$; if $n$ is odd, the sharp radius is strictly smaller for $r\in(0,1)$ and depends on $...
We establish o-minimal definability results and develop point-counting machinery applicable to the zeros of a Bessel function of fixed non-negative real parameter. We also give an Ax-Schanuel theorem for Bessel functions of fixed (complex) parameters and exponential functions. We combine these to prove a sparsity resul...
Gal Binyamini, Avner Kiro, G. Kozma et al.· 0 citations
We resolve a conjecture on the law of large numbers for coherent measures on Young diagrams associated with Macdonald-nonnegative specializations of the algebra of symmetric functions for all $0\le q,t<1$. We prove that row and column lengths divided by the diagram size converge to the corresponding specialization para...
Vadim Gorin· 0 citations
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