论文标题
三角多项主义具有频率在一组正方形中
Trigonometric polynomials with frequencies in the set of squares
论文作者
论文摘要
令$γ_0= \ frac {\ sqrt5-1} {2} = 0.618 \ ldots $。我们证明，对于任何$ \ varepsilon> 0 $和任何三角多项式$ f $ f $，在集合中具有频率的$ \ {n^2：n \ leqslant n \ leqslant n \ leqslant n+n+n+n+n+n+n+n+n+n+n+n+n+n+n+ \ varepsilon^{ - 1/4} \ | f \ | _2 $$保持，这在Cilleruelo和Cordoba的猜想上取得了进展。我们还提出了这种猜想与Ruzsa的猜想之间的联系，该猜想断言，对于任何$ \ varepsilon> 0 $，都有$ c（\ varepsilon）> 0 $，因此每个正整数$ n $最多都有$ c（\ varepsilon）$ c（\ varepsilon）$ divisors $在此间隔$ [n^n^n^n^n^^{1/2 {1/2 {1/1/2}， n^{1/2}+n^{1/2- \ varepsilon}] $
Let $γ_0=\frac{\sqrt5-1}{2}=0.618\ldots$ . We prove that, for any $\varepsilon>0$ and any trigonometric polynomial $f$ with frequencies in the set $\{n^2: N \leqslant n\leqslant N+N^{γ_0-\varepsilon}\}$, the inequality $$ \|f\|_4 \ll \varepsilon^{-1/4}\|f\|_2 $$ holds, which makes a progress on a conjecture of Cilleruelo and Cordoba. We also present a connection between this conjecture and the conjecture of Ruzsa which asserts that, for any $\varepsilon>0$, there is $C(\varepsilon)>0$ such that each positive integer $N$ has at most $C(\varepsilon)$ divisors in the interval $[N^{1/2}, N^{1/2}+N^{1/2-\varepsilon}]$
