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One thing you can try to use to get estimates for $\mathcal T$ is bounds for the number of divisors $2^k\leq d\leq 2^{k+1}$. Since pairs of divisors from different dyadic ranges have distinct sums, yo …

Shouldn't this happen as long as $\mathbf{a}$ and $\mathbf{b}$ are linearly dependent mod $p$? For if so, you are talking about points in $\mathbb{Z}^n$ which reduce mod $p$ in two some $n-2$ dimensio …

In an abelian group, the additive energy between two sets is $$E(A,B)=|\{(a_1,a_2,b_1,b_2) \in A\times A\times B\times B:a_1+b_1=a_2+b_2\}|$$ which is ranges from $|A||B|$ to $(|A||B|)^{3/2}$. What I …

Suppose $f$ and $g$ are polynomials with integral coefficients and $f(\mathbb Z)\subset g(\mathbb Z)$. Is there any relation between $f$ and $g$? For instance, this happens if $f=g\circ h$ for some …

It is a fairly well known fact that sparse polynomials $f(x)$ cannot have large order zeros other than at $x=0$. If $f(x)=a_1x^{r_1}+\cdots+a_kx^{r_k}$ then at $c \neq 0$, $f$ has a zero of order at m …

Suppose $a$ is a square-free integer and $\left(\frac{a}{p}\right)=1$ for the primes $p\leq k$. I'll call $a$ a quasi-square of order $k$. What I am interested in is the maximum value of $k$ in terms …

Suppose $A\subset[1,N]$ is a set of integers. If for any distinct $a,b\in A$ we have $(a,b)\leq M$ then how big can $|A|$ be? If $M=1$ then $|A|$ is at most $\pi(N)$ since the map $a\mapsto P_+(a)$ …

Suppose $p,q$ are distinct primes with least quadratic non-residues $n_p$ and $n_q$ respectively. Can one bound the least $n$ for which $\left(\frac{n}{p}\right)=\left(\frac{n}{q}\right)=-1$ in terms …

Suppose $F(x,y)$ is a polynomial in two variables over a field $K$, and $F(x,y)$ is not a square. When is $F(x,a)$ a square for $a\in K$? I would guess that Hilbert's Irreducibility Theorem might help …

Well, certainly $L=\text{lcm}(1,...,n)$ is divisible by $p^k\leq n$. We can take $k$ up to $[\log_p n]\geq \frac{\log n}{\log p}-1$. So $$\log L\geq \sum_{p\leq n}\left(\frac{\log n}{\log p}-1\right)\ …