Additive problems with smooth integers

2016 ◽  
Author(s):  
H. Ki ◽  
H. Maier ◽  
A. Sankaranarayanan
Keyword(s):  
Smooth Integers

scholarly journals Large strings of consecutive smooth integers

2011 ◽  
Vol 97 (4) ◽  
pp. 319-324
Author(s):  
Filip Najman
Keyword(s):  
Smooth Integers

scholarly journals The multiplication table for smooth integers

2021 ◽  
Vol 219 ◽  
pp. 172-197
Author(s):  
Marzieh Mehdizadeh
Keyword(s):  
Multiplication Table ◽  
Smooth Integers

scholarly journals Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers

2016 ◽  
Vol 152 (6) ◽  
pp. 1121-1158 ◽  
Author(s):  
Adam J. Harper
Keyword(s):  
Riemann Hypothesis ◽  
Exponential Sums ◽  
Upper Bounds ◽  
Weighted Sums ◽  
Number Of Solutions ◽  
Mean Values ◽  
Smooth Numbers ◽  
Prime Factors ◽  
Moment Bounds ◽  
Smooth Integers

We investigate exponential sums over those numbers ${\leqslant}x$ all of whose prime factors are ${\leqslant}y$. We prove fairly good minor arc estimates, valid whenever $\log ^{3}x\leqslant y\leqslant x^{1/3}$. Then we prove sharp upper bounds for the $p$th moment of (possibly weighted) sums, for any real $p>2$ and $\log ^{C(p)}x\leqslant y\leqslant x$. Our proof develops an argument of Bourgain, showing that this can succeed without strong major arc information, and roughly speaking it would give sharp moment bounds and restriction estimates for any set sufficiently factorable relative to its density. By combining our bounds with major arc estimates of Drappeau, we obtain an asymptotic for the number of solutions of $a+b=c$ in $y$-smooth integers less than $x$ whenever $\log ^{C}x\leqslant y\leqslant x$. Previously this was only known assuming the generalised Riemann hypothesis. Combining them with transference machinery of Green, we prove Roth’s theorem for subsets of the $y$-smooth numbers whenever $\log ^{C}x\leqslant y\leqslant x$. This provides a deterministic set, of size ${\approx}x^{1-c}$, inside which Roth’s theorem holds.

Smoothing ‘smooth’ numbers

1993 ◽  
Vol 345 (1676) ◽  
pp. 339-347 ◽  
Keyword(s):  
Smooth Numbers ◽  
Short Intervals ◽  
Prime Factors ◽  
Fixed Power ◽  
Smooth Integers

An integer is called y-smooth if all of its prime factors are ⩽ y . An important problem is to show that the y -smooth integers up to x are equi-distributed among short intervals. In particular, for many applications we would like to know that if y is an arbitrarily small, fixed power of x then all intervals of length x up to x , contain, asymptotically, the same number of y -smooth integers. We come close to this objective by proving that such y -smooth integers are so equi-distributed in intervals of length x y 2 + ε , for any fixed ε < 0.

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