Allen-Cahn equation for the truncated Laplacian: Unusual phenomena

1.   Introduction

Diophantine equation is a classical problem in number theory. Let $[\alpha]$ denote the integer part of the real number $\alpha$ and $N$ be a sufficiently large integer. In 1933, Segal ^[27,28] firstly studied additive problems with non-integer degrees, and proved that there exists a $k_0(c) > 0$ such that the Diophantine equation

is solvable for $k > k_0(c)$, where $c > 1$ is not an integer. Later, Deshouillers ^[5] improved Segal's bound of $k_0(c)$ to $6c^3(\log c+14)$ with $c > 12$. Further Arkhilov and Zhitkov ^[1] refined Deshouillers's result to $22c^2(\log c+4)$ with $c > 12$. Afterwards, many results of various Diophantine equations were established (e.g., see ^{[7,10,14,16,17,18,19,21,25,41,42]}). In particular, Laporta ^[17] in 1999 showed that the equation

is solvable in primes $p_1$, $p_2$ provided that $1 < c < \frac{17}{16}$ and $N$ is sufficiently large. Recently, the range of $c$ in (1.2) was enlarged to $1 < c < \frac{14}{11}$ by Zhu ^[40]. Kumchev ^[15] showed that the equation

is solved for almost all $N$ provided that $1 < c < \frac{16}{15}$, where $m$ is an integer and $p$ is a prime. Afterwards, the range of $c$ in (1.3) was enlarged to $1 < c < \frac{17}{11}$ by Balanzario, Garaev and Zuazua ^[3].

In 1995, Laporta and Tolev ^[18] considered the equation

with prime variables $p_1, p_2, p_3$. Denote the weighted number of solutions of Eq (1.4) by

They established the following asymptotic formula

for any $0 < \delta < \frac{1}{3}$ and $1 < c < \frac{17}{16}$. Afterwards, the range of $c$ was enlarged to $1 < c < \frac{12}{11}$ by Kumchev and Nedeva ^[16], to $1 < c < \frac{258}{235}$ by Zhai and Cao ^[39], and to $1 < c < \frac{137}{119}$ by Cai ^[4].

In this paper, we first show a more general result related to (1.5) by proving the following theorem.

Theorem 1.1. Let $N$ be a sufficiently large integer. Then for $1 < c < \frac{3+3\kappa-\lambda}{3\kappa+2}$, we have

for any $0 < \delta < \frac{1}{3}$, where $(\kappa, \lambda)$ is an exponent pair, and the implied constant in the $O-$symbol depends only on $c$.

Choosing $(\kappa, \lambda) = BA^2BABABABAB(0, 1) = \left(\frac{81}{242}, \frac{132}{242}\right)$ in Theorem 1.1, we can immediately get the following corollary, which further improves the result of Cai ^[4].

Corollary 1.2. Under the notations of Theorem 1.1, for $1 < c < \frac{837}{727}$ the asymptotic formula $(1.6)$ follows.

It is easy to verify that the range of $c$ in Corollary 1.2 is larger than one of Cai's result. Our improvement mainly derives from more accurate estimates of exponential sums by combining Van der Corput's method, exponent pairs and some elementary methods. Also the estimates of exponential sums has lots of applications in problems including automorphic forms (e.g., see ^{[8,11,12,20,22,24,29,30,31,32,33,34,35,36,37,38]}).

Notation. Throughout the paper, $N$ always denotes a sufficiently large integer. The letter $p$, with or without subscripts, is always reserved for primes. Let $\varepsilon\in\Big(0, 10^{-10}(\frac{3+3\kappa-\lambda}{3\kappa+2}-c)\Big)$. We denote by $\{x\}$ and $\|x\|$ the fraction part of $x$ and the distance from $x$ to the nearest integer, respectively. Let $1 < c < \frac{3+3\kappa-\lambda}{3\kappa+2}$ and

2.   Auxiliary lemmas

To prove Theorem 1.1, we need the following lemmas.

Lemma 2.1 ([9,Lemma 5]). Suppose that $z_n$ is a sequence of complex numbers, then we have

where ${\rm Re}(t)$ and $\overline{t}$ denote the real part and the conjugate of the complex number $t$, respectively.

Lemma 2.2. Suppose that $|x| > 0$ and $c > 1$. Then for any exponent pair $(\kappa, \lambda)$ and $M\leq a < b\leq2M$, we have

Proof. We can get this lemma from [6,(3.3.4)].

Lemma 2.3 ([2,Lemma 12]). Suppose that $t$ is not an integer and $H\geq3$. Then for any $\alpha\in(0, 1)$, we have

where

Lemma 2.4 ([9,Lemma 3]). Suppose that $3 < U < V < Z < X$, and $\{Z\} = \frac{1}{2}, \ X\geq64Z^2U, \ Z\geq 4U^2, \ V^3\geq32X$. Further suppose that $F(n)$ is a complex valued function such that $|F(n)|\leq1$. Then the sum

can be decomposed into $O(\log^{10}X)$ sums, each of which either of type {I}:

with $N > Z$, where $a(m) \ll m^{\varepsilon}$ and $X\ll MN\ll X$, or of type {II}:

with $U\ll M\ll V$, where $a(m) \ll m^{\varepsilon}, b(n) \ll n^{\varepsilon}$ and $X\ll MN\ll X$.

Lemma 2.5. Let $f(t)$ be a real value function and continuous differentiable at least three times on $[a, b] \; (1\leq a < b\leq2a)$, $|f'''(x)|\sim\Delta > 0$, then we have

Moreover, if $0 < c_1\lambda_1\leq c_2\lambda_1$, $|f''(x)|\sim\lambda_1a^{-1}$, then we have

if $c_2\lambda_1\leq\frac{1}{2}$, then we have

Proof. The first result was proved by Sargos ^[26]. And the remaining two results were due to Jia ^[13].

Lemma 2.6 ([23,Lemma 2]). Let $M > 0$, $N > 0$, $u_m > 0$, $\upsilon_n > 0$, $A_m > 0$, $B_n > 0\ (1\leq m\leq M, 1\leq n\leq N)$. Let also $Q_1$ and $Q_2$ be given non-negative numbers, $Q_1\leq Q_2$. Then there is one $q$ such that $Q_1\leq q\leq Q_2$ and

Lemma 2.7 ([36,Lemma 5]). Let $f(x)$, $g(x)$ be algebraic functions in $[a, b]$, $|f''(x)|\sim\frac{1}{R}$, $f'''(x)\ll\frac{1}{RU}$, $U\geq1$, $|g(x)|\ll G$, $|g'(x)|\ll GU^{-1}$. $[\alpha, \beta]$ is the image of $[a, b]$ under the mapping $y = f'(x)$. $n_u$ is the solution of $f'(n) = u$.

Then we have

Lemma 2.8 ([13,Lemma 3]). Suppose that $x\sim N$, $f(x)\ll P$, and $f'(x)\gg\Delta$. Then we have

Lemma 2.9. For $0 < \alpha < 1$ and any exponent pair $(\kappa, \lambda)$, we have

Proof. Throughout the proof of this lemma, we write $H = X^{\frac{-\kappa c+1-\lambda+\kappa}{1+\kappa}}$ for convenience. Using Lemma 2.3 we can get

Then by the expansion

where

we have

where we estimated the sum over $n$ by Lemma 2.2.

In a similar way, we have

Then this lemma follows.

Lemma 2.10 ([42,Lemma 2.1]). Suppose that $f(n)$ is a real-valued function in the interval $[N, N_1]$, where $2\leq N < N_1 \leq 2N$. If $0 < c_1 \lambda_1 \leq |f'(n)| \leq c_2\lambda_1 \leq \frac{1}{2},$ then we have

If $|f^{(j)}(n)|\sim \lambda_1 N^{-j+1} \; (j = 1, 2)$, then we have

If $|f^{(j)}(n)|\sim \lambda_1 N^{-j+1} \; (j = 1, 2, 3, 4, 5, 6)$, then we have

where $(\kappa, \lambda)$ is any exponent pair.

Lemma 2.11 ([9,Lemma 6]). Suppose that $0 < a < b\leq 2a$ and $R$ is an open convex set in $\mathbb{C}$ containing the real segment $[a, b]$. Suppose further that $f(z)$ is analytic on $R$. $f(x)$ is real for real $x\in R$. $f''(z)\leq M$ for $z\in R$. There is a constant $k > 0$ such that $f''(x)\leq -k M$ for all real $x\in R$. Let $f'(b) = \alpha$ and $\ f'(a) = \beta$, and define $x_{\upsilon}$ for each integer $\upsilon$ in the range $\alpha < \upsilon < \beta$ by $f'(x_{\upsilon}) = \upsilon$. Then we have

3.   The estimate of $S(\alpha)$

Lemma 3.1. Let $P^{\frac{5}{6}}\ll X\ll P$, $H = X^{1-\frac{(1+2\kappa)c+\lambda}{2+2\kappa}}$ and $c_h(\alpha)$ denote complex numbers such that $c_h(\alpha)\ll(1+|h|)^{-1}$. Then uniformly for $\alpha\in(\tau, 1-\tau)$, we have

for any $a(m)\ll m^\varepsilon$, where $(\kappa, \lambda)$ is any exponent pair, $X\ll MN\ll X$ and $M\ll Y$ with $Y = \min\left\{X_1, X_2, X_3, X_4, X_5, X_6, X_7, X_8\right\}$,

Proof. It is easy to deduce that

where $K_h = \sum_{m\sim M}\left|\sum_{n\sim N}e\left((\alpha+h)(mn)^c\right)\right|$. According to Hölder's inequality, we have

Let $z_n = z_n(m, \alpha) = (\alpha+h)(mn)^c$. Suppose that $Q$, $J$ are two positive integers such that $1\leq Q\leq N\log^{-1}X$, $1\leq J\leq N\log^{-1}X$. For the inner sum in (3.2), applying Lemma 2.1 twice, we can get

where

Let $\Delta(n^c; q, j) = (n+q+j)^c-(n+q)^c-(n+j)^c+n^c$, $G(m, n) = (\alpha+h) m^c\Delta(n^c; q, j)$. Then $z_n-z_{n+q}-z_{n+j}+z_{n+q+j} = G(m, n)$. Thus we have

For any $t\neq1, 0$, we have

then

and

If $c(c-1)(c-2)(\alpha+h)qjM^c N^{c-3}\leq\frac{1}{100}$, by Lemma 2.5 we have

From now we always suppose that $c(c-1)(c-2)(\alpha+h)qjM^c N^{c-3}\geq\frac{1}{100}$. By Lemma 2.7 we have

where

By Lemma 2.8, the contribution of $R(m, q, j)$ to $E(q, j)$ is

Then we only need to deal with the following exponential sum

where $I_\upsilon$ is a subinterval of $[M, 2M]$. For a fixed $\upsilon$, we define $\Delta_{\lambda'} = \Delta\left(n_\upsilon^{\lambda'}; q, j\right)$, where $\lambda'$ is an arbitrary real number. We take the derivative of $m$ in (3.8) and get

It follows from (3.7) that

Recalling (3.6), we can get

Thus for $m$, $\left|\frac{\partial^2G}{\partial n^2}(m, n_\upsilon)\right|^{-\frac{1}{2}}$ is monotonic. Let $g(m) = G(m, n_\upsilon(m))-\upsilon n_\upsilon(m)$. Then we have

where

From the above formulas we can obtain

Using Lemma 2.5 and partial summation we can get

By (3.5), (3.9) and (3.11), we have

Inserting (3.12) into (3.3), we obtain

Then choosing optimal $J\in[0, N\log^{-1}X]$ and $Q\in[0, N\log^{-1}X]$ and using Lemma 6 twice we can get

where

Recalling the definitions of $H$ and $Y$, we have

and Lemma 3.1 is proved.

Lemma 3.2. Let $P^{\frac{5}{6}}\ll X\ll P$, $H = X^{1-\frac{(1+2\kappa)c+\lambda}{2+2\kappa}}$, $F = (h+\alpha)X^c$and $c_h(\alpha)$ denote complex numbers such that $c_h(\alpha)\ll(1+|h|)^{-1}$. Then uniformly for $\alpha\in(\tau, 1-\tau)$, we have

for any $a(m)\ll m^\varepsilon$, $b(n)\ll n^\varepsilon$, where $(\kappa, \lambda)$ is any exponent pair, $X\ll MN\ll X$ and

Proof. Taking $Q = [X^{2-\frac{(1+2\kappa)c+\lambda}{1+\kappa}}\log^{-1}X]$, then we have $Q = o(N)$. By Cauchy's inequality and Lemma 2.1, we have

where $E_q = \sum_{n\sim N}|b(n+q)b(n)|\left|\sum_{m\sim M}e(G(mn))\right|$ and $G(m, n) = G(m, n, q) = (h+\alpha)m^c\Delta(n, q;c)$, $\Delta(n, q;c) = (n+q)^c-n^c$.

If $|\frac{\partial G}{\partial m}| \leq 10^3Mq^{-2}$, by Lemma 2.10 we have

noting that $M\gg\frac{X}{F}$.

Now we suppose $|\partial G/\partial m| > 10^3Mq^{-2}$. By Lemma 11 we get

where $s(r, n) = G(M(r, n), n)-rm(r, n), \ \varphi(r, n) = \frac{(Fq)^\frac{1}{2}}{MN^\frac{1}{2}}\left|\frac{\partial^2 G(m(r, n), n)}{\partial m^2}\right|^{-\frac{1}{2}}$ and

Thus we have

So it suffices to bound the sum

Let $T = [Fq^3/M^2N]$ and $R = Fq/MN$. By Cauchy's inequality and Lemma 2.1 again we get

where

and where we used the estimate

It is easy to check that $10 < T = o(R)$.

Recall that $s(r, n) = G(m(r, n), n)-rm(r, n)$, where $m(r, n)$ denotes the solution of

It is easy to deduce that

So we can obtain

which implies that $|H^{(j)}|\sim tMN^{-j}$, $(j = 0, 1, 2, 3, 4, 5, 6)$. Denote by $I(r, t)$ the interval $N < n\leq 2N$, $r_1 (n) < n\leq r_2 (n)-t$. Then we have

Thus using partial summation, we get

with the exponent pair $(\kappa, \lambda)$, if we note that $M\gg X^{26-\frac{53}{3}\frac{(1+2\kappa)c+\lambda}{1+\kappa}}F^{\frac{53}{9}}$. From (3.15)–(3.18) we get that for any $1\leq q \leq Q$,

Now this lemma follows from inserting (3.19) into (3.14).

Lemma 3.3. For $\tau\leq\alpha\leq1-\tau$, we have

where $(\kappa, \lambda)$ is any exponent pair.

Proof. Throughout the proof of this lemma, we write $H = X^{1-\frac{(1+2\kappa)c+\lambda}{2+2\kappa}}$ for convenience. By a dissection argument we only need to prove that

holds for $P^{\frac{5}{6}}\leq X\leq P$ and $\tau\leq\alpha\leq1-\tau$. According to Lemma 2.3, we have

By the expansion

where

we get

where we estimated the sum over $n$ by Lemma 2.2 with the exponent pair $\left(\frac{1}{2}, \frac{1}{2}\right)$.

Let $R = \max\left\{X^{3-\frac{(1+2\kappa)c+\lambda}{1+\kappa}}F^{-1}, X^{4-\frac{2(1+2\kappa)c+2\lambda}{1+\kappa}}, X^{26-\frac{53}{3}\frac{(1+2\kappa)c+\lambda}{1+\kappa}}F^{\frac{53}{9}}\right\}$. Recall the definition of $Y$ in Lemma 3.1. Let $U = R$, $V = X^{{\frac{(1+2\kappa)c+\lambda}{1+\kappa}}-1}$, $Z = [XY^{-1}]+\frac{1}{2}$. By Lemma 4 with $F(n) = e((h+\alpha)n^c)$, then we reduce the estimate of

to the estimates of sums of type ${I}$

and sums of type ${II}$

where $a(m)\ll m^\varepsilon, \ b(n)\ll n^\varepsilon, \ X\ll MN \ll X$. By Lemma 3.1, we get

By Lemma 3.2, we get

From (3.23) and (3.24) we can obtain

Now (3.20) follows from (3.21), (3.22) and (3.25). Thus we complete the proof of this Lemma.

4.   Proof of the theorem

It is easy to see that

4.1.   Evaluation of $R_1(N)$

Following the argument of Laporta and Tolev [18,pages 928–929], we can get that

for $1 < c < \frac{3}{2}$ and any $0 < \delta < \frac{1}{3}$, where the implied constant in the $O-$symbol depends only on $c$.

4.2.   Evaluation of $R_2(N)$

Let

We can get

where we used

Now, we start to estimate the absolute value on the right hand in (4.3). By Cauchy's inequality we have

Then we have

By Lemma 3.3, we have

where we used the trivial bound $T(\alpha, X)\ll X$. By Lemma 2.9, Lemma 3.3 and (4.4), we get

Thus, combining (4.6)–(4.8) we obtain

By (4.3), (4.5) and (4.9), we can obtain

Now putting (4.1), (4.2) and (4.10) into together, we have

follows for any $0 < \delta < \frac{1}{3}$, where the implied constant in the $O-$symbol depends only on $c$. Thus we complete the proof of Theorem 1.1.

Acknowledgment

The authors would like to thank the referees for their many useful comments. This work is supported by National Natural Science Foundation of China (Grant Nos. 11801328 and 11771256).

Conflict of interest

The authors declare no conflict of interest.