arXiv subjects

Naser T Sardari

Publications and source records attributed to Naser T Sardari.

4 recordsLinked to original sources

arXiv2018

Complexity of strong approximation on the sphere

By assuming some widely-believed arithmetic conjectures, we show that the task of accepting a number that is representable as a sum of $d\geq2$ squares subjected to given congruence conditions is NP-complete. On the other hand, we develop and implement a deterministic polynomial-time algorithm that represents a number as a sum of 4 squares with some restricted congruence conditions, by assuming a polynomial-time algorithm for factoring integers and Conjecture~\ref{cc}. As an application, we develop and implement a deterministic polynomial-time algorithm for navigating LPS Ramanujan graphs, under the same assumptions.

math.NT↗

arXiv2018

Optimal strong approximation for quadratic forms

For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in $d\geq5$ variables, we prove an optimal strong approximation theorem. Let $Ω$ be a fixed compact subset of the affine quadric $F(x_1,\dots,x_d)=1$ over the real numbers. Take a small ball $B$ of radius $0 0$. Finally assume that an integral vector $(λ_1, \dots, λ_d) $ mod $m$ is given. Then we show that there exists an integral solution $X=(x_1,\dots,x_d)$ of $F(X)=N$ such that $x_i\equiv λ_i \text{ mod } m$ and $\frac{X}{\sqrt{N}}\in B$, provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a non-degenerate integral quadratic form in 4 variables we prove the same result if $N$ is odd and $N\gg_{δ,Ω} (r^{-1}m)^{6+ε}$. Based on our numerical experiments on the diameter of LPS Ramanujan graphs and the expected square root cancellation in a particular sum that appears in Remark~\ref{evidence}, we conjecture that the theorem holds for any quadratic form in 4 variables with the optimal exponent $4$.

math.NT↗

arXiv2017

Diameter of Ramanujan Graphs and Random Cayley Graphs

We study the diameter of LPS Ramanujan graphs $X_{p,q}$. We show that the diameter of the bipartite Ramanujan graphs is greater than $ (4/3)\log_{p}(n) +O(1)$ where $n$ is the number of vertices of $X_{p,q}$. We also construct an infinite family of $(p+1)$-regular LPS Ramanujan graphs $X_{p,m}$ such that the diameter of these graphs is greater than or equal to $ \lfloor (4/3)\log_{p}(n) \rfloor$. On the other hand, for any $k$-regular Ramanujan graph we show that the distance of only a tiny fraction of all pairs of vertices is greater than $(1+ε)\log_{k-1}(n)$. We also have some numerical experiments for LPS Ramanujan graphs and random Cayley graphs which suggest that the diameters are asymptotically $(4/3)\log_{k-1}(n)$ and $\log_{k-1}(n)$, respectively.

math.NT↗

arXiv2017

Quadratic forms and semiclassical eigenfunction hypothesis for flat tori

Let $Q(X)$ be any integral primitive positive definite quadratic form with discriminant $D$ and in $k$ variables where $k\geq4$. We give an upper bound on the number of integral solutions of $Q(X)=n$ for any integer $n$ in terms of $n$, $k$ and $D$. As a corollary, we give a definite answer to a conjecture of Rudnick and Lester on the small scale equidistribution of orthonormal basis of eigenfunctions restricted to an individual eigenspace on the flat torus $\mathbb{T}^d$ for $d\geq 5$. Another application of our main theorem gives a sharp upper bound on $A_{d}(n,t)$, the number of representation of the positive definite quadratic form $Q(x,y)=nx^2+2txy+ny^2$ as a sum of squares of $d\geq 5$ linear forms where $n- n^{\frac{1}{(d-1)}-o(1)}< t < n$. This upper bound allows us to study the local statistics of integral points on sphere.

math.NT↗