For a vector of positive integers $\mathbf{b} = (b_1,\ldots,b_h)$ with $\gcd(b_1,\ldots,b_h) = 1$, we study sets $A \subseteq \mathbb{N}$ for which every sufficiently large integer has a bounded positive number of representations \[ n = b_1 x_1 + \cdots + b_h x_h \qquad (x_1,\ldots,x_h\in A). \] We prove that such a set exists for every binary vector $\mathbf{b} \neq (1,1)$, and for some general higher-dimensional families, including $\mathbf{b} = (u_1, p^d u_2, \ldots, p^{(h-1)d} u_h)$ where $p\nmid u_1\cdots u_h$.
Let $F=(f_1(x_1,\ldots,x_k),\ldots, f_m(x_1,\ldots,x_k))$ be a system of nonconstant polynomials of $k$ variables with integer coefficients and let \[ {\gcd}_F(x_1,\ldots,x_k)= \gcd(f_1(x_1,\ldots,x_k),\ldots, f_m(x_1,\ldots,x_k)). \] We obtain an unconditional asymptotic formula for the sum \[ \sum_{1\le x_1,\ldots,x_...
Let $P=\{0,a,b\}$, where $0<a<b$ and $\gcd(a,b)=1$. For a finite set $A\subset\mathbb Z$, let $M_P^+(A)$ count the copies $x,x+ad,x+bd\in A$ with $d>0$, and let $M_P(A)$ count the copies with any $d\ne0$. We prove that every such three-point pattern other than the arithmetic progression $\{0,1,2\}$ satisfies \[ M_P^+(A...
For an integer $d\ge 3$, put $\Delta_d=\min{2^{d-1},d(d-1)}$. Let $a/q$ be reduced, let $P(X)=\frac{a}{q}X^d+\alpha_{d-1}X^{d-1}+\cdots+\alpha_0$, and let $\mathcal{I}$ be an interval of $H\le q$ consecutive integers. We prove $\left|\sum_{n\in\mathcal{I}}e(P(n))\right|\ll_{d,\varepsilon}q^{1/d}H^\varepsilon+H^{1-1/\De...
Let $b_1,\dots,b_d\ge2$ be fixed integers, and let $k$ be their multiplicative rank. We study representations $n=a+b_1^{u_1}\cdots b_d^{u_d}$, where $a$ belongs to a set $\mathcal{A}$ of positive integers and $u_1,\dots,u_d$ are positive integers, counting distinct tuples of exponents separately. Under density and corr...
Let $p_1,\dots,p_r\geq 5$ be distinct primes, let $P=p_1\cdots p_r$, and put $M=6P$. For a positive integer $n$, let $g_P(2n)$ denote the number of unordered representations $2n=h+k$, with $1\leq h\leq k$, such that $\gcd(h,M)=\gcd(k,M)=1$. Using the canonical remainder operator $\delta_q(x)=x-q\lfloor x/q\rfloor$, we...
We study random polynomials of the form $R(x)=x^n+\omega_{n-1}x^{n-1}+\cdots+\omega_0$, where $\omega_0,\dots,\omega_{n-1}$ are independent, uniformly bounded integer-valued random variables, and $\omega_1,\dots,\omega_{n-1}$ have a fixed common law $\mu$. We prove (unconditionally) that, if the R\'{e}nyi entropy of or...
Guy Blachar, E. Breuillard, G. Kozma· 0 citations
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