Additive energies on spheres
Abstract: In this paper, we study additive properties of finite sets of lattice points on spheres in three and four dimensions. Thus, given d,m∈double-struckN$d,m \in \mathbb {N}$, let A$A$ be a set of lattice points (x1,⋯,xd)∈Zd$(x_1, \dots , x_d) \in \mathbb {Z}^d$ satisfying x12+⋯+xd2=m$x_1^2 + \dots + x_{d}^2 = m$. When d=4$d=4$, we prove threshold breaking bounds for the additive energy of A$A$, that is, we show that there are at most Oε(mεfalse|Afalse|2+1/3−1/2766)$O_{\epsilon }(m^{\epsilon }|A|^{2 + 1/3 - 1/2766}… Show more
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Cited by 2 publications
References 19 publications
Given $$h,g \in {\mathbb {N}}$$ h , g ∈ N , we write a set $$X \subset {\mathbb {Z}}$$ X ⊂ Z to be a $$B_{h}^{+}[g]$$ B h + [ g ] set if for any $$n \in {\mathbb {Z}}$$ n ∈ Z , the number of solutions to the additive equation $$n = x_1 + \dots + x_h$$ n = x 1 + ⋯ + x h with $$x_1, \dots , x_h \in X$$ x 1 , ⋯ , x h ∈ X is at most g, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative $$B_{h}^{\times }[g]$$ B h × [ g ] set analogously. In this paper, we prove, amongst other results, that there exist absolute constants $$g \in {\mathbb {N}}$$ g ∈ N and $$\delta >0$$ δ > 0 such that for any $$h \in {\mathbb {N}}$$ h ∈ N and for any finite set A of integers, the largest $$B_{h}^{+}[g]$$ B h + [ g ] set B inside A and the largest $$B_{h}^{\times }[g]$$ B h × [ g ] set C inside A satisfy $$\begin{aligned} \max \{ |B|, |C| \} \gg _{h} |A|^{(1+ \delta )/h }. \end{aligned}$$ max { | B | , | C | } ≫ h | A | ( 1 + δ ) / h . In fact, when $$h=2$$ h = 2 , we may set $$g = 31$$ g = 31 , and when h is sufficiently large, we may set $$g = 1$$ g = 1 and $$\delta \gg (\log \log h)^{1/2 - o(1)}$$ δ ≫ ( log log h ) 1 / 2 - o ( 1 ) . The former makes progress towards a recent conjecture of Klurman–Pohoata and quantitatively strengthens previous work of Shkredov.
Given $$h,g \in {\mathbb {N}}$$ h , g ∈ N , we write a set $$X \subset {\mathbb {Z}}$$ X ⊂ Z to be a $$B_{h}^{+}[g]$$ B h + [ g ] set if for any $$n \in {\mathbb {Z}}$$ n ∈ Z , the number of solutions to the additive equation $$n = x_1 + \dots + x_h$$ n = x 1 + ⋯ + x h with $$x_1, \dots , x_h \in X$$ x 1 , ⋯ , x h ∈ X is at most g, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative $$B_{h}^{\times }[g]$$ B h × [ g ] set analogously. In this paper, we prove, amongst other results, that there exist absolute constants $$g \in {\mathbb {N}}$$ g ∈ N and $$\delta >0$$ δ > 0 such that for any $$h \in {\mathbb {N}}$$ h ∈ N and for any finite set A of integers, the largest $$B_{h}^{+}[g]$$ B h + [ g ] set B inside A and the largest $$B_{h}^{\times }[g]$$ B h × [ g ] set C inside A satisfy $$\begin{aligned} \max \{ |B|, |C| \} \gg _{h} |A|^{(1+ \delta )/h }. \end{aligned}$$ max { | B | , | C | } ≫ h | A | ( 1 + δ ) / h . In fact, when $$h=2$$ h = 2 , we may set $$g = 31$$ g = 31 , and when h is sufficiently large, we may set $$g = 1$$ g = 1 and $$\delta \gg (\log \log h)^{1/2 - o(1)}$$ δ ≫ ( log log h ) 1 / 2 - o ( 1 ) . The former makes progress towards a recent conjecture of Klurman–Pohoata and quantitatively strengthens previous work of Shkredov.
A quadratic Vinogradov mean value theorem in finite fields
Let p be a prime, let $s \geq 3$ be a natural number and let $A \subseteq \mathbb{F}_p$ be a non-empty set satisfying $|A| \ll p^{1/2}$. Denoting $J_s(A)$ to be the number of solutions to the system of equations $$ \sum_{i=1}^{s} (x_i - x_{i+s}) = \sum_{i=1}^{s} (x_i^2 - x_{i+s}^2) = 0, $$ with $x_1, \dots, x_{2s} \in A$, our main result implies that $$ J_s(A) \ll |A|^{2s - 2 - 1/9}. $$ This can be seen as a finite field analogue of the quadratic Vinogradov mean value theorem. Our techniques involve a variety of combinatorial geometric estimates, including studying incidences between Cartesian products A × A and a special family of modular hyperbolae.
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